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# option leverage ratio

However, most analysts consider that UPS earns enough cash to cover its debts. The most commonly used leverage ratio is the debt to equity ratio expressed as follows: Debt to Equity Ratio = Total Liabilities/Total Shareholders Equity. * By submitting your email address, you consent to receive email messages (including discounts and newsletters) regarding Corporate Finance Institute and its products and services and other matters (including the products and services of Corporate Finance Institute's affiliates and other organizations). This ratio indicates that the higher the degree of financial leverage, the more volatile earnings will be. This will help frame up the risk at 10-93x option leverage. Leverage allows you to gain more exposure with less money. Generally, leverage ratio above 2 is considered high in debt burden, and a harder time to recover during economic downturn due to high interest expenses. Banking regulations for leverage ratios are complicated. Remember that Total Assets = Total Debt + Total shareholders' Equity. Derivatives clearing, bank leverage, and options trades make a potent concoction. But as you know leverage can be good or bad depending on how you use it. Exploration costs are typically found in the financial statements as exploration, abandonment, and dry hole costs. If the option does not reach the required price by expiration though the option will expire worthless, which means you lose the premium paid. Leverage ratio squeeze hits options trades With clearing banks constrained by leverage limits, prop traders fear options market lockdown. It's a good idea to measure a firm's leverage ratios against past performance and with companies operating in the same industry to better understand the data. Print this page . Such a standard seeks to ensure that the exposures and commitments of a bank do not exceed a certain multiple of its capital, irrespective of the level of risk associated with them. It is calculated as: Total debt to capitalization=(SD+LD)(SD+LD+SE)where:SD=short-term debtLD=long-term debtSE=shareholders’ equity\begin{aligned} &\text{Total debt to capitalization} = \frac{(SD + LD)}{(SD + LD + SE)}\\ &\textbf{where:}\\ &SD=\text{short-term debt}\\ &LD=\text{long-term debt}\\ &SE=\text{shareholders' equity}\\ \end{aligned}Total debt to capitalization=(SD+LD+SE)(SD+LD)where:SD=short-term debtLD=long-term debtSE=shareholders’ equity. If your trade goes horribly wrong, the most you can lose is the premium amount (As they say - limited risk, infinite profit potential). The option to temporarily relax the leverage ratio during the pandemic was introduced by . There is a familiar story that pops up every so often in the pages of the financial press: “Option Strategy Blows Up, Loses Nearly All of Investors’ Money.” Readers shake their heads and wonder how anyone could be so foolish to lose 50%, 75% or 100% so quickly. In this case, if the Margin-Based Leverage Expressed Ratio is 1:100 than the Margin Required of Total Transaction Value will be 1.00%. By using Investopedia, you accept our, Investopedia requires writers to use primary sources to support their work. Save this article. There are several different leverage ratios that may be considered by market analysts, investors, or lenders. So 1:100 leverage is the best leverage to be used in forex trading. The equity multiplier is similar, but replaces debt with assets in the numerator: Equity Multiplier=Total AssetsTotal Equity\text{Equity Multiplier} = \frac{\text{Total Assets}}{\text{Total Equity}}Equity Multiplier=Total EquityTotal Assets. Options leverage is a double edged sword which returns the same multiple on both option trading profits and losses. Degree of financial leverage (DFL) is a ratio that measures the sensitivity of a company’s earnings per share (EPS) to fluctuations in its operating income, as a result of changes in its capital structure. An operating leverage ratio refers to the percentage or ratio of fixed costs to variable costs. A l’inverse, il sera attendu d’une entreprise positionnée sur un secteur d’activité cyclique ou dont la conjoncture est incertaine un ratio d’endettement net plus faible. Brokers set their rates, which in some cases can reach 1:100 or even more. When an individual deals with options, futures, margins, or other financial instruments. Formula, examples. Below is an illustration of two common leverage ratios: debt/equity and debt/capital. Unlike margin, leveraged ETFs, or options, this is an indirect method of adding leverage to your portfolio. Your leverage ratio will vary, depending on the market you are trading, who you are trading it with, and the size of your position. Browse hundreds of articles! These statements are key to both financial modeling and accounting. • a credible leverage ratio is one that ensures broad and adequate capture of both the on- and off-balance sheet sources of banks’ leverage. You can learn more about the standards we follow in producing accurate, unbiased content in our. Interest Coverage Ratio (ICR) is a financial ratio that is used to determine the ability of a company to pay the interest on its outstanding debt. Because the leverage ratio is highly sensitive to the notional value of a position, it is much more Improved BDC Leverage Options Set… Jan 13 2020. The higher the Options Leverage multiple, the higher the potential reward and the higher the risk of loss. Instead of using long-term debt, an analyst may decide to use total debt to measure the debt used in a firm's capital structure. Les produits à effet de levier (warrants et turbos) offrent la possibilité de se positionner à la hausse (Call) ou à la baisse (Put) sur un actif sous-jacent, avec un effet de levier. The equity multiplier would be: $19.85 billion÷$4.32 billion=4.59\$19.85 \text{ billion} \div \$4.32 \text{ billion} = 4.59$19.85 billion÷$4.32 billion=4.59. BDCs are well-established as an essential provider of debt capital to small and medium-sized U.S. businesses. For example, assume that Macy's (NYSE: M) has assets valued at $19.85 billion and stockholder equity of$4.32 billion. If the ratio of fixed costs to revenue is high (i.e., >50%)  the company has significant operating leverage. It's important to have an understanding of these important terms. Facebook . If the relevant Put or Call Options are Exercised you will be required to trade a multiple of the Notional Amount of the contract at an Above that level, debt costs increase considerably. Options Leverage =. Leverage ratios Leverage Ratios A leverage ratio indicates the level of debt incurred by a business entity against several other accounts in its balance sheet, income statement, or cash flow statement. Besides the ratios mentioned above, we can also use the coverage ratiosCoverage RatioA Coverage Ratio is used to measure a company’s ability to pay its financial obligations. Investopedia uses cookies to provide you with a great user experience. A Cash Flow Statement (officially called the Statement of Cash Flows) contains information on how much cash a company has generated and used during a given period. A leverage ratio may also be used to measure a company's mix of operating expenses to get an idea of how changes in output will affect operating income. Without leverage, traders will able to earn a very low percentage of profits like .3% to .5%. Margin-Based Leverage Ratio = Total Value of Transaction / Margin Required. Leverage ratios represent the extent to which a business is utilizing borrowed money. Leverage ratio squeeze hits options trades With clearing banks constrained by leverage limits, prop traders fear options market lockdown. An example of a capital-intensive business is an automobile manufacturing company. Leverage is an important feature offered by Forex brokers. However there is also additional risk associated with the Leverage Ratio. Before investing money, it is necessary to check an appropriate leverage ratio. So before your next option buy convert the delta to a leverage ratio. Commonly used by credit agencies, this ratio determines the probability of defaulting on issued debt. Banks have regulatory oversight on the level of leverage they are can hold. The combination of fractional-reserve banking and Federal Deposit Insurance Corporation (FDIC) protection has produced a banking environment with limited lending risks. Leverage is one of the key concepts and benefits of options trading. Explosive Short Term Option Trading Find Out How My … Leverage can thus multiply returns, although it can also magnify losses if returns turn out to be negative. On the other hand, high financial leverage ratios occur when the return on investment (ROI) does not exceed the interest paid on loans. "2019 Annual Report," Page 2. Leverage Ratio a Leveraged Structured Option provides an enhanced Exchange Rate in comparison to the equivalent standard Structured Option. United Parcel Service's total stockholders' equity for the ending December 2019 was $3.3 billion. A leverage ratio is any one of several financial measurements that assesses the ability of a company to meet its financial obligations. When comparing debt to equity, the ratio for this firm is 0.82, meaning equity makes up a majority of the firm’s assets. The leverage ratio a ects low (absolute) delta options more than high (absolute) delta options. For banks, the tier 1 leverage ratio is most commonly used by regulators. In this guide, we will break down the EV/EBTIDA multiple into its various components, and walk you through how to calculate it step by step, This financial modeling guide covers Excel tips and best practices on assumptions, drivers, forecasting, linking the three statements, DCF analysis, more, Join 350,600+ students who work for companies like Amazon, J.P. Morgan, and Ferrari, Certified Banking & Credit Analyst (CBCA)™, Capital Markets & Securities Analyst (CMSA)™, certified financial analyst training program, Financial Modeling & Valuation Analyst (FMVA)®,$2 million of annual depreciation expense, A company takes on debt to purchase specific assets. Current Leverage: (0.3858/2.22)*293.74 = 51.04 This option has over 51x leverage. The Income Statement is one of a company's core financial statements that shows their profit and loss over a period of time. Use pre-tax earnings because interest is tax-deductible; the full amount of earnings can eventually be used to pay interest. An indicator that measures the amount of debt in a company’s capital structure is the debt-to-capitalization ratio, which measures a company’s financial leverage. Large US banks must hold 3%. A leverage ratio is any one of several financial measurements that assesses the ability of a company to meet its financial obligations. Finally, analyzing the existing level of debt is an important factor that creditors consider when a firm wishes to apply for further borrowing. FDIC. IV. Disclaimer - Use of OptionPlus involves the use of financial leverage. The calculation is simple: current leverage = (delta/option price)*current stock price. Glossary of terms and definitions for common financial analysis ratios terms. Ce ratio en particulier est spécifiquement conçu pour le secteur bancaire et il permet de mesurer la solidité d’un établissement financier. Options with low delta have low market value and low market risk relative to a comparable notional of high-delta options. The equity multiplier is a calculation of how much of a company’s assets is financed by stock rather than debt. Send to . Download the free Excel template now to advance your finance knowledge! The profit or loss is determined by taking all revenues and subtracting all expenses from both operating and non-operating activities.This statement is one of three statements used in both corporate finance (including financial modeling) and accounting. This is usually a type of “cash flow loan” and is generally only available to larger companies. If the company's interest expense grows too high, it may increase the company's chances of a default or bankruptcy. This leads investors to conclude options are dangerous instruments that should be avoided altogether.But this line of reasoning is far too simplistic.There is nothing fundamentally dangerous or risky about options. If the ratio of fixed costs to revenue is low (i.e., <20%) the company has little operating leverage. Concernant les entreprises « classiques », il existe d’autres ratios. It measures the percentage change in EPS for a unit change in earnings before interest and taxes (EBIT) and is represented as: DFL=% change in EPS% change in EBITwhere:EPS=earnings per shareEBIT=earnings before interest and taxes\begin{aligned} &DFL = \frac{\% \text{ change in }EPS}{\% \text{ change in }EBIT} \\ &\textbf{where:}\\ &EPS=\text{earnings per share}\\ &EBIT=\text{earnings before interest and taxes}\\ \end{aligned}DFL=% change in EBIT% change in EPSwhere:EPS=earnings per shareEBIT=earnings before interest and taxes. This ratio (100% / 20%) is the options leverage. Les Leverage et Short n’ont pas de barrière désactivante, ils présentent toutefois un risque de perte partiel ou total de capital en cas de variation importante du sous-jacent. Times interest earned (TIE), also known as a fixed-charge coverage ratio, is a variation of the interest coverage ratio. To calculate this ratio, find the company's earnings before interest and taxes (EBIT), then divide by the interest expense of long-term debts. Use the below OptionPlus Leverage calculator to compare the margin requirements for trading options with OptionPlus and without OptionPlus. There are several forms of capital requirements and minimum reserve placed on American banks through the FDIC and the Comptroller of the Currency that indirectly impacts leverage ratios. One problem with only reviewing the total debt liabilities for a company is they do not tell you anything about the company's ability to service the debt. The oil industry seems to have about a 40% debt-to-capital threshold. A leverage ratio may also be used to measure a company's mix of operating expenses to get an idea of how changes in output will affect operating income. A high debt/equity ratio generally indicates that a company has been aggressive in financing its growth with debt. The schedule should outline all the major pieces of debt a company has on its balance sheet, and calculate interest by multiplying the. Typically, a D/E ratio greater than 2.0 indicates a risky scenario for an investor; however, this yardstick can vary by industry. Assets financed by debt cannot be written down because the bank's bondholders and depositors are owed those funds. Too much debt can be dangerous for a company and its investors. The third type of leverage ratio relates to consumer debt, which is compared to disposable income. Generally, a ratio of 3.0 or higher is desirable, although this varies from industry to industry. Options offer the ability to customize stock profiting strategies using leverage. Estimating option leverage is a helpful way to help understand the risk/reward of an option position. The Basel leverage ratio has caused banks to reduce their clearing activity for E S&P 500 E-mini futures. They adds that there are various measures known for the capital structure among which the most important are book value based measures, market value based measures and semi- market value based measures (adjusted market value). That is considered high.. The formula to calculate options leverage is defined as: Options Leverage (or Lambda) = Delta x Stock price (S) / Option price (V) with Delta = dV / dS = Change of option price / Change of stock price In our example: Options Leverage = 0.5 x $20 /$2.0 = 5 The profit or loss is determined by taking all revenues and subtracting all expenses from both operating and non-operating activities.This statement is one of three statements used in both corporate finance (including financial modeling) and accounting., or cash flow statementCash Flow StatementA Cash Flow Statement (officially called the Statement of Cash Flows) contains information on how much cash a company has generated and used during a given period. Excessive use of financing can lead to default and bankruptcy. The balance sheet is one of the three fundamental financial statements. The fixed-charge coverage ratio (CFFR) indicates a firm's capacity to satisfy fixed charges, such as debt payments, insurance premiums, and equipment leases. Having high leverage in a firm’s capital structure can be risky, but it also provides benefits. In accordance with the CRR, institutions have to report to their supervisors all necessary information on the leverage ratio and its components. Recap: Delta is an estimate of how the value of an option changes for each $1 move in the underlying stock. Leverage of 1:100 means that with$500 in the account, the trader has $50,000 of credit funds provided by the broker to open trades. The debt-to-capital ratio is a measurement of a company's financial leverage. Until late 2018, BDCs were permitted to incur debt up to a 1:1 ratio to the fair market value of their respective portfolios. L’un des plus connus est le ratio : Dette nette / Fonds propres, qui permet de mesurer le niveau d’endettement. Save this article. Print this page . So if the stock goes up 10% — your$200 option bet will go up by an even larger amount– say, 50%, depending on which option you choose. The debt to asset ratio, also known as the debt ratio, is a leverage ratio that indicates the percentage of assets that are being financed with debt. Having both high operating and financial leverage ratios can be very risky for a business. Fedex has a D/E ratio of 1.78, so there is cause for concern where UPS is concerned. Uncontrolled debt levels can lead to credit downgrades or worse. Using the example from earlier, a 10% margin would provide the same exposure as a $1000 investment with just$100 margin.

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text:
# What are the two numbers that are the sum 50 difference 10 ? thank you

May 25, 2018

See below.

#### Explanation:

Firstly, assign the two numbers random variables $x$ and $y$

The sum of them is equal to $50$ therefore

$x + y = 50$

The difference is $10$

$x - y = 10$

Now we have a simultaneous equation.
$x + y = 50$
$x - y = 10$

Add them together to cancel out the $y$.

$2 x = 60$

Now solve for $x$ $\implies x = 30$

Now put the value back into one of the equations to find $y$

$y + 30 = 50$
$\implies y = 20$

The two numbers are $30$ and $20$

May 25, 2018

$30 \text{ and } 20$

#### Explanation:

"let the 2 numbers be x and y ";x>y

$x + y = 50 \leftarrow \textcolor{b l u e}{\text{sum of numbers}}$

$x - y = 10 \leftarrow \textcolor{b l u e}{\text{difference of numbers}}$

$\text{add the 2 equations term by term on both sides}$

$\left(x + x\right) + \left(y - y\right) = \left(50 + 10\right)$

$2 x = 60$

$\text{divide both sides by 2}$

$x = \frac{60}{2} = 30 \Rightarrow x = 30$

$\text{substitute " x=30" into } x + y = 50$

$30 + y = 50$

$\text{subtract 30 from both sides}$

$y = 50 - 30 = 20 \Rightarrow y = 20$

$\text{the 2 numbers are 30 and 20}$

May 25, 2018

30 and 20

#### Explanation:

Okay let's define a couple numbers, let's call one of them $x$ and the other $y$.

We are told that the sum (addition) is:

$x + y = 50$

And the difference (subtraction):

$x - y = 10$

We have a system of equations; two equations and two unknown variables so it is solvable; we will use the "substitution" method:

add $y$ to both sides of: $x - y = 10$

$x - y + y = 10 + y$

$x = 10 + y$

now substitute the value we solved for $x$ into the other equation:

$x + y = 50$

$\left(10 + y\right) + y = 50$

$10 + 2 y = 50$

$2 y = 40$

$y = 20$

So one of the numbers is $20$. to find the other use either of our original equations and insert $y$ to solve for $x$, this one is simplest:

$x + y = 50$

$x + 20 = 50$

$x = 30$

Solved! Our numbers are 30 and 20

To check your solutions insert them into the original equations:

$x + y = 50$

$30 + 20 = 50$

and

$x - y = 10$

$30 - 20 = 10$

format: plain


text:
# How do you solve abs(2/3x-9)=18?

May 19, 2017

$x = - \frac{27}{2}$
$x = + \frac{81}{2}$

#### Explanation:

For this to work we have to end up with $| \pm 18 | = + 18$

This means that $\text{ } \frac{2}{3} x - 9 = \pm 18$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Case 1: $\text{ } \frac{2}{3} x - 9 = + 18$

$\frac{2}{3} x = 27$

Multiply both sides by $\frac{3}{2}$

$x = \frac{3}{2} \times 27 = 40 \frac{1}{2} \to \frac{81}{2}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Case 2: $\text{ } \frac{2}{3} x - 9 = - 18$

$\frac{2}{3} x = - 9$

$x = \frac{3}{2} \times \left(- 9\right) = - \frac{27}{2}$


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# M61-18

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18 Jun 2018, 04:04
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If $$2^n+2^{n-2}=5120$$, then $$n=?$$

A. 8
B. 9
C. 10
D. 11
E. 12

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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6529 GMAT 1: 760 Q51 V42 GPA: 3.82 Re M61-18 [#permalink] ### Show Tags 18 Jun 2018, 04:04 Official Solution: If $$2^n+2^{n-2}=5120$$, then $$n=?$$ A. 8 B. 9 C. 10 D. 11 E. 12 Factoring yields $$2^{n}+2^{n-2}=2^{2}2^{n-2}+2^{n-2}=(2^2+1)2^{n-2}=5*2^{n-2}=5120=5*1024$$. Therefore, $$2^{n-2}=1024=2^{10}$$ and $$n-2 = 10$$. It follows that $$n = 12$$. Answer: E _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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14 Sep 2018, 05:08
MathRevolution wrote:
If $$2^n+2^{n-2}=5120$$, then $$n=?$$

A. 8
B. 9
C. 10
D. 11
E. 12

Hello Can anyone please explain this question in a simpler manner? it would be of great help. Thank you!
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25 Oct 2018, 03:43
I agree with Shri15kumar. The explanation is not clear. I got the answer correct but only because I guessed and I would really like to learn how to solve a problem like this. Thank you!
Re: M61-18 &nbs [#permalink] 25 Oct 2018, 03:43
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# M61-18

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# Andrew and Stephen drive on the highway in the same direction at respe

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Andrew and Stephen drive on the highway in the same direction at respe  [#permalink]

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Andrew and Stephen drive on the highway in the same direction at respective rates of 72 and 80 kmh. If Stephen is 4 km behind Andrew, by how much does he have to increase his speed to catch up with Andrew in 20 minutes?

A. 1 kmh
B. 2 kmh
C. 3 kmh
D. 4 kmh
E. 5 kmh

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Re: Andrew and Stephen drive on the highway in the same direction at respe  [#permalink]

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19 Oct 2014, 02:58
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Andrew and Stephen drive on the highway in the same direction at respective rates of 72 and 80 kmh. If Stephen is 4 km behind Andrew, by how much does he have to increase his speed to catch up with Andrew in 20 minutes?

A. 1 kmh
B. 2 kmh
C. 3 kmh
D. 4 kmh
E. 5 kmh

M17-37

Denote the required acceleration as $$x$$. The distance between Andrew and Stephen will be decreasing at $$(80 + x - 72)$$ kmh. We can compose the equation $$\frac{4}{80 + x - 72} = \frac{20}{60}$$ from which $$x = 4$$.

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Re: Andrew and Stephen drive on the highway in the same direction at respe  [#permalink]

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25 Jul 2009, 13:42
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I think its much easier to first assume that they are going the same speed. So he has to catch up 4 meters in 1/3 hour. r= d/t = 4/(1/3) = 12 km/hr faster he has to go. He is already going 8km/h faster, so 12 - 8 = 4 km/hr
##### General Discussion
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Re: Andrew and Stephen drive on the highway in the same direction at respe  [#permalink]

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22 Apr 2009, 18:44
2
4
A -- 72 Kmh
B -- 80 Kmh

Relative speed = 8 kmh

distance b/w them = 4 Km

4 /( 80 +x - 72) = 20/60

solve for x = 4 .....

Ans . D
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Re: Andrew and Stephen drive on the highway in the same direction at respe  [#permalink]

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02 May 2009, 07:36
6
3
If x is the distance travelled by Andrew in 20 minutes = 72*20/60 = 24km

Stephen's speed would to travel (24+4)km = 28*60/20 = 84 km/h

So, answer is D - 4km/h
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Re: Andrew and Stephen drive on the highway in the same direction at respe  [#permalink]

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02 May 2009, 15:25
2
2
relative speed=80-72=8
need to catch up 4km

currently Stephen is catching up at a speed of 4km/30mins, or 2km/15mins
he needs to increase the speed, so that he can make 4km in 20 mins instead, or 1km/5mins

x - the relative speed increase we seek
2/15+x=1/5
x=1/15

So, he needs to increase relative speed by 1km per 15mins, or 4km/h
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Re: Andrew and Stephen drive on the highway in the same direction at respe  [#permalink]

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12 Nov 2015, 20:04
4k/8kph=1/2h
4k/(8+x)kph=1/3h
x=4kph faster
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Re: Andrew and Stephen drive on the highway in the same direction at respe  [#permalink]

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03 Feb 2017, 12:47
2
2
Andrew and Stephen drive on the highway in the same direction at respective rates of 72 and 80 kmh. If Stephen is 4 km behind Andrew, by how much does he have to increase his speed to catch up with Andrew in 20 minutes?

A. 1 kmh
B. 2 kmh
C. 3 kmh
D. 4 kmh
E. 5 kmh

We can classify this problem as a “catch-up” rate problem, for which we use the formula:

distance of Andrew + 4 = distance of Stephen

We are given that Andrew is driving at a rate of 72 kmh and that Stephen is driving at a rate of 80 kmh. We need to determine how much faster Stephen needs to drive to catch up to Andrew in 20 minutes or 20/60 = 1/3 hour.

If we let r = Stephen’s speed increase, we can say the following:

distance of Stephen = (r + 80)(1/3) = (r + 80)/3

distance of Andrew + 4 = 72 x 1/3 + 4 = 28

We can equate the two distances and determine r.

(r + 80)/3 = 28

r + 80 = 84

r = 4

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Re: Andrew and Stephen drive on the highway in the same direction at respe  [#permalink]

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03 Nov 2017, 11:17
1
Andrew and Stephen drive on the highway in the same direction at respective rates of 72 and 80 kmh. If Stephen is 4 km behind Andrew, by how much does he have to increase his speed to catch up with Andrew in 20 minutes?

A. 1 kmh
B. 2 kmh
C. 3 kmh
D. 4 kmh
E. 5 kmh

M17-37

given
4/8 = 1/2

to make it 1/3, we need to change 8 to 12
so the difference is 4

thanks
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Re: Andrew and Stephen drive on the highway in the same direction at respe  [#permalink]

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04 Nov 2017, 21:35
Relative speed = 8 kmh

distance b/w them = 4 Km

4 /( 80 +x - 72) = 20/60

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Re: Andrew and Stephen drive on the highway in the same direction at respe  [#permalink]

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03 Dec 2018, 03:41
at current speeds,

current time to catch up= 4/8 = 1/2; where 8 is the relative speed and 1/2 means 30 minutes
we want: 4/x = 1/3; where 1/3 means 20 minutes
x=12

s - 72 = 12 --> s=84 --> inc by 4
Re: Andrew and Stephen drive on the highway in the same direction at respe   [#permalink] 03 Dec 2018, 03:41
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3 added 91 characters in body

As I suggested in my short comment, this kind of question has been around for a long time and has led to a vast amount of literature. It probably starts with work over fields by Schreier and van der Waerden in the 1920s, then considerable work by Dieudonne, O'Meara, and many others. Two indications about what's out there are a survey by O'Meara The integral classical groups and their automorphisms (1969) and a short paper by me dealing from the algebraic group viewpoint with groups like $SL_n(\mathbb{Q})$, On the automorphisms of infinite Chevalley groups, Canad. J. Math. 21 1969 908–911, the latter probably available online by now.

These are listed on MathSciNet, but I couldn't display the links there for some reason. I'll try to suggest more focused literature on your question when I get time today.

P.S. Ed Formanek has pointed to a basic early paper by Hua and Reiner here. Like many other papers on related automorphism groups, the emphasis is on identification of special types of automorphisms which suffice to generate the whole group: inner, "field" or "ring" (as in complex conjugation and the like), "graph" (as in transpose-inverse map for special linear groups), "diagonal" (as in the use of conjugation by diagonal matrices not of determinant 1 to produce automorphisms of a subgroup). Naturally the ring or field automorphisms play no role over the rational integers or rational numbers.

2 added 589 characters in body

As I suggested in my short comment, this kind of question has been around for a long time and has led to a vast amount of literature. It probably starts with work over fields by Schreier and van der Waerden in the 1920s, then considerable work by Dieudonne, O'Meara, and many others. Two indications about what's out there are a survey by O'Meara The integral classical groups and their automorphisms (1969) and a short paper by me dealing from the algebraic group viewpoint with groups like $SL_n(\mathbb{Q})$, On the automorphisms of infinite Chevalley groups, Canad. J. Math. 21 1969 908–911, the latter probably available online by now.

These are listed on MathSciNet, but I couldn't display the links there for some reason. I'll try to suggest more focused literature on your question when I get time today.

P.S. Ed Formanek has pointed to a basic early paper. Like many other papers on related automorphism groups, the emphasis is on identification of special types of automorphisms which suffice to generate the whole group: inner, "field" or "ring" (as in complex conjugation and the like), "graph" (as in transpose-inverse map for special linear groups), "diagonal" (as in the use of conjugation by diagonal matrices not of determinant 1 to produce automorphisms of a subgroup). Naturally the ring or field automorphisms play no role over the rational integers or rational numbers.

1

As I suggested in my short comment, this kind of question has been around for a long time and has led to a vast amount of literature. It probably starts with work over fields by Schreier and van der Waerden in the 1920s, then considerable work by Dieudonne, O'Meara, and many others. Two indications about what's out there are a survey by O'Meara The integral classical groups and their automorphisms (1969) and a short paper by me dealing from the algebraic group viewpoint with groups like $SL_n(\mathbb{Q})$, On the automorphisms of infinite Chevalley groups, Canad. J. Math. 21 1969 908–911, the latter probably available online by now.

These are listed on MathSciNet, but I couldn't display the links there for some reason. I'll try to suggest more focused literature on your question when I get time today.


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# Force-torque sensor reading when a robot is stationary

Let's say I have a robot arm with a force-torque sensor on the wrist, between the final motor and the gripper. If the arm and wrist are stationary, then what would the reading on the sensor be?

I know this might sound like a silly question, but I don't understand which of the following would be true:

• The reading would be g (gravity), since the only force acting on the sensor is gravity.
• The reading would be g + w, where w is the weight of the gripper, because both gravity and the gripper's weight are acting on the sensor.
• The reading would be 0, because whilst g + w are acting down on the sensor, since the arm is stationary, there must be an equal an opposite upward force applied by the robot, which exactly cancels out the downwards force.

Which of these is correct? And consequently, why are the other two not correct?

Thanks!

## 1 Answer

The answer is $$\vec{G}$$! and also any $$\vec{\tau}$$ torque caused by it, if the axis connecting the geometrical center of the sensor (assuming it measures torques relative to its geometrical center) to the centre of gravity of the gripper.

$$\vec{G} = m * \vec{g}$$

where $$m$$ is the mass of the gripper, $$g$$ is the gravitational ACCELERATION, not force and G is the gravitational pull (force) on the gripper.

• Thank you! So, ignoring torque created by gravity, we can say that the reading will be m * g. Forgive me if this is a silly question, but isn't there also an upwards force coming from the arm to hold the gripper in place, which should be equal to m * g? Why wouldn't this cancel out the force of gravity, such that the sensor reads 0? – Karnivaurus Mar 1 at 20:50
• No. If the sensor reads 0 it is in freefall, as in there is no force and no reaction force. Imagine the sensor as a membrane. The gravitational pull is pulling the middle of the membrane down and the base (side) is kept in place by the reaction force. You are right, action an reaction cancel out, but exactly action and reaction allows us to measure. If you hold a kitchen scale in your hand and press it against your other hand, you can measure the force. If you do not press it against anything and exert a force on it it will move and you cannot measure the force. – 50k4 Mar 1 at 21:53
• Ok, thank you! Newton's laws confused me with these kind of things... if every action has an equal and opposite reaction, how is there ever any acceleration...?! Anyway, I will have a think about this! – Karnivaurus Mar 1 at 23:02


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Question

# Fluorine exhibits only $$-1$$ oxidation state where as other halogens exhibit $$+1,+3,+5$$ and $$+7$$ oxidation states also. Explain.

Solution

## Fluorine is the most electronegative element and cannot exhibit any positive oxidation state as it does not have d orbitals in its valence shell. Other halogens have $$d-orbitals$$ and therefore, can expand their octets and show $$+1, +3, +5$$ and $$+7$$ oxidation states also.Chemistry

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Question: setdiff for GenomicRanges
0
21 months ago by
Brazil/Campinas/UNICAMP
Benilton Carvalho4.2k wrote:

Hi everyone,

I'm wondering if the following is the expected behaviour for setdiff:

gr1 <- GRanges('a', IRanges(c(1, 3), c(2, 9)))
gr2 <- GRanges('a', IRanges(20, 30))
gr3 <- GRanges('a', IRanges(c(1, 4), c(2, 9)))
diff1 <- setdiff(gr1, gr2)
diff2 <- setdiff(gr3, gr2)

My expectation was to get gr1 back, given that the intersection between gr1 and gr2 is empty. But the resulting object diff1 is reduce(gr1). Just to be clear, I expected to get something analogous to diff2.

Many thanks, benilton

modified 21 months ago by Michael Lawrence10.0k • written 21 months ago by Benilton Carvalho4.2k
3
21 months ago by
Michael Lawrence10.0k
United States
Michael Lawrence10.0k wrote:

Yes, this is the expected behavior, because this is a set operation, and all set operations imply reduce(). We view the ranges as sets of integers, and sets must only contain unique elements.

To subtract all ranges in one set (B) that overlap each range in another (A), find the overlaps, group the B ranges by overlap, and use psetdiff():

hits <- findOverlaps(gr1, gr2)
grl <- extractList(gr2, as(hits, "List"))
psetdiff(gr1, grl)

But note that you get back a GRangesList, since ranges in A can be split. You'll need to think about how to deal with those.


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# Project: Photographic methods of making PCBs

#### Lestraveled

Joined May 19, 2014
1,946
First, I would like to pay homage to Wendy Marsden and her contribution of her thread on making PCBs with the toner transfer method. It is a lot of work to put together a thread like that. In that thread, she said she would like to see someone start a thread on the photographic method of making PCBs and thus, I was motivated to create this tread. Thank you Wendy.

This thread is about making PCBs using photographic processes beginning with transparencies (artwork) produced from a laser or inkjet printer. There is a lot of flexibility when you choose the photographic path. So, this thread is not about one way of making PCBs photographically, it is about all the ways to make PCBs photographically. Based on your set of conditions and preferences, there are a variety of tools and options that you can use to optimize your PCB fabrication. Each process has its benefits and its costs.

Many of us in this community have a lot of experience making our own PCBs and we have our own ways of how to do it. I think it would be truly interesting and beneficial to hear about how the other guy makes their boards and especially, why they do it that way. I know that I have wrestled with making PCBs for a long time, and some of those times, the PCB has won.

So, I will start by presenting how I make PCBs and why I do it that way. Afterwards, I will take each function and expand on it, so that you can, if you want to, use it in your PCB fabrication.

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#### Lestraveled

Joined May 19, 2014
1,946
The process outlined below is the result of years of messing up a lot of PCBs. Frustration and wasting money are great motivators for me. As a result I have honed my process down to the essentials that guarantee me that when I finish, at least 90% of the PCBs I make are in the excellent category, and that I have not spent a lot of money doing it. I get no satisfaction from working several hours, burning through some money and then coming out of the shop with a handful of ratty looking PCBs. So, if a step, like turning my artwork into litho film, will significantly improve the quality and yield, I will spend the extra time and effort to insure that I am happy with the fruits of my labor.

Even though I have used a lot of them, I am not a fan of pre-sensitized positive PCBs. I think they are too expensive and too inconsistent. Also, you cannot tell if the exposure was correct until you put it in the etchant. At that point, if the exposure was not correct, then the board is not reusable. I hate wasting money. I am sure that I have messed up enough pre-sensitized positive boards to pay for the laminator that I now use to apply negative resist film to PCB stock. At minimum, this topic should inspire a lively discussion.

Outline of my process (Details of all of the processes are in later posts)

I print 3 or 4 positives of the PCB image on laser transparency film. (My laser printer does not print very dark.) I then stack/align them together to increase the density of the black image. The following pictures are of a single transparency and of 4 transparencies aligned.

Single transparency

Four Transparencies aligned on top of one another

I then use Litho film to reverse the image and to increase the contrast. The benefits of using Litho film, as well as how to use it, are covered in a later post.

Once the artwork is ready, it is time to sensitize the PCB stock. I use negative acting dry film resist. This film is a photo sensitive emulsion sandwiched between two protective sheets of plastic. One side of the plastic is removed and a laminator is used to apply it to the PCB stock. Dry Film resist has proven itself to be more reliable and precise than pre-sensitized positive PCB stock, plus it is much much cheaper. Again, this is covered in detail, with a video, in a later post.

It is now time to expose the PCB. I use a contact print frame to hold everything together while exposing. I use a 5 watt UV LED (365nm) as my light source. Once exposed, the image is clearly visible on the PCB. This is a nice feature of dry film resist. If there is a problem, the board can be stripped and re-laminated in just a few minutes.

The second layer of plastic strips off easily and the board is developed in a solution of Sodium Carbonate and water.

Development complete

I etch my boards in Sodium Persulfate using an economy etcher. Using this etcher has greatly reduced my consumption of sodium persulfate. I can etch three times the number of boards for the same amount of sodium persulfate as compared to using an unheated tray. It is a lot faster also.

The finished board (on a light box)

While the above process might seem a bit advanced for a home hobbyist, it works very well and is very reliable. If you are making more than one board, then it works even better. Good artwork is the key to success regardless of the photo resist you are using. You may not be interested in going through all of the above steps, but I am certain that you will find something of value in this thread that will improve the success rate, and or reduce the cost, of your PCB process.

There are videos in later posts that detail all of the above actions.

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#### Lestraveled

Joined May 19, 2014
1,946
Laser transparencies – What you see is not what you get – Something you need to know

This applies to both positive and negative acting resists

My HP1160 laser printer does not print very dark on transparencies. I have set the print density to 5, econo mode to off, and tried HP and aftermarket cartridges. A grainy dark gray is as dense as it gets, so I have to print 3 or 4 transparencies and align them on top of one another (stack up) to get the density/contrast up. Then I use the stack up as the master for the litho film. In the next picture there is a positive litho film on the left and the stack up on the right.

They both look very good....... in visible light, but PCB resist does not expose in visible light, it responds to a narrow band of UV light. In other words, what you see in visible light, is not what you get in ultra violet light. A significant factor is that plastic attenuates UV light a lot. Every plastic transparency I add to improve the density of the black image, reduces the transparency of the clear image at ultra violet. With every layer of plastic I add, the entire artwork gets darker at UV and contrast is lost. This throws off the exposure times and shrinks the optimal exposure time window (the sweet spot).

It is a little difficult to demonstrate this effect. We cannot see 365nm wavelength UV light and I don’t have UV light meter. We can demonstrate this indirectly by observing how much a piece of white paper fluoresces due to the UV light striking it.

The following picture needs some explanation. On the bench is a white sheet of paper lying flat. 12 inches above the paper, is an ultra violet light source (365nm) shining down. There is a plastic box on opposite sides of the paper. On top of each box is a transparency that extends over the paper. The two transparencies are arranged so that they cast a shadow on the paper. The art work on the left is a single layer litho film. The art work of the right is four layers of laser transparency film. The same PCB image is on both films.

Notice the shadow of the litho film on the left. It is bright and has high contrast. Compare it to the shadow on the right. The shadow on the right has reduced intensity and reduced contrast. This is due to it having four layers of plastic. These two pieces of artwork looked almost identical in visible light, and as you can see, they are very different at Ultra Violet. The reduced intensity directly impacts your exposure time. The reduced contrast shrinks the time difference between a board being under exposed and over exposed.

Bottom line: If your laser printer prints a very dark dense image and you only need one or two layers of transparencies to get good density, then you probably can safely bypass using Litho film. If your laser printer is crappy, like mine, then consider adding a litho film step in your process. You will not regret it. Turning my artwork into litho film has had a huge impact on the quality and yield of my PCB making.

Last edited:

#### Lestraveled

Joined May 19, 2014
1,946
Artwork and exposing

This applies to both positive and negative acting resists

The art work is the photographic pattern used to control where the copper is going to remain on the PCB stock. With positive artwork, the image is black where the copper will stay, and with negative artwork, the image is clear where the copper will stay. The pattern is transferred to the sensitized PCB by a process called “contact printing”. The artwork is laid directly on top of the sensitized PCB, and a UV light source shines through the artwork and on to the photo resist. With positive resist, the exposed areas soften and will be washed away when developed. With negative resist, the exposed areas will harden and remain when developed.

During the exposure, the artwork must be pressed firmly against the surface of the PCB. This is normally done with a “contact printing frame (or easel)”. The construction of a printing frame is fairly simple. From the bottom up, you start with a flat solid base such as Plywood, MDF, ¼” glass etc. (Flatness is very important.) Next is a sheet of foam rubber, ¼” is good. Next is your sensitized PCB, resist side up. Next is the artwork. Then finally a sheet of glass, ¼” is good. Binding clips, rubber bands, or spring clips hold the “sandwich” together. The sandwich is then placed under the UV light source and exposed. You can also purchase a photographic contact printing frame.

You will see the above printing frame in use in later videos.

Off Contact Exposure
There is a condition called “off contact exposure”. This is when there is a gap between the artwork and the PCB resist (off contact) and the exposing light undercuts the artwork pattern. This can result in cut traces when positive resist is used and shorted traces when negative resist is used. Off contact exposure is caused by several things; not enough pressure on the “sandwich”, a warped PCB or base. A major cause of off contact exposure is wrinkles in the artwork, especially if layered laser transparencies are used. A laser printer uses heat to melt the toner on to the plastic transparency. This heat distorts the plastic transparency, thus creating wrinkles and possibly off contact exposures. Off contact exposure can be minimized by a single point UV light source, discussed in a later post.

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#### Lestraveled

Joined May 19, 2014
1,946
PCB Exposure sources

This applies to both positive and negative acting resists

Each photo resist has a specific UV wavelength that it works best at for exposing, but all of the wavelengths are in the UV “A” range, 315 to 400 nm. UV "A" fluorescent bulbs are common, and available from the hardware store. I currently use a 365 nm. LED to expose dry film resist. I have not tested a 365nm LED on positive resist, but I suspect it would work just fine.

You should check the spec on exposing your photo resist before choosing a light source.

Here are few common light sources that produce these wavelengths:
- The sun – cheap but can be inconsistent
- Fluorescent UV "A" bulbs (tubes) – Medium cost, bad for off contact exposure
- Photographic incandescent bulbs – Low/medium cost, hot,
- UV LEDs (bed of nails) - Medium cost, bad for off contact exposure
- UV LED (single, Hi Pwr. diode) – Medium/low cost, reduces off contact exposure

If there are any wrinkles or perturbations in the art work film, off contact exposure can occur due to gaps between the artwork and the PCB. The shape of the light source effects how badly these gaps affect the PCB traces.

The following shows the advantages of a point light source.

Below is a picture of a “bed of tubes” UV exposer and it does a good job if there are no gaps between the artwork and the PCB.

Below is a picture of the shadow that a pencil casts using the “bed of tubes” exposer placed 12 inches away from a piece of white paper. Notice that the pencil casts almost no shadow due to the multiple light sources (tubes). This means that if there is a gap between the artwork and the PCB, that gap will be exposed and will undercut the shape of the trace.

Below is a “point” light source made from a single 5 watt UV (365nm) LED.

Below is a picture of the shadow that the pencil casts using the “point” light source. The shadow is clear and sharp. This means that if there is a gap between the artwork and the PCB there will be little of no undercutting of the PCB pattern.

Bottom line: If you use laser transparencies for your exposure artwork, then you would greatly benefit from using a single high power UV LED to expose with. I bought my diode for $17, power it with a 12V wall wart and use a 10 ohm resistor to limit the current. It is mounted on an old CPU heat sink. It is a lot cheaper and works better than a$100 fluorescent exposer.

My homemade point UV light source.

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#### Lestraveled

Joined May 19, 2014
1,946
Ortho Litho Film

This applies to both positive and negative acting resists

The reason for using ortho litho flim is that it greatly improves the quality, contrast and UV opaqueness of your artwork. ("Ortho" just means that you can handle it in subdued red light.)

Litho film is photographic film designed to have only two states, clear or opaque black, and no grey tones. Think of this film as having a threshold. If the amount of light striking the film is less than the threshold then the film stays completely clear. If the amount of light is over the threshold then the film goes to full black. This feature is how the film turns images that are not so dense, like those generated by my crappy printer, into extremely high contrast, extremely dense artwork. Another feature of litho film is that it is exposed with visible light, yet when developed, the black areas are UV opaque and the clear areas attenuate UV light very little.

Ortho litho film comes in sheets or rolls. You handle it with a red safe light. To develop it you need a developer liquid, a fixer liquid and water.

Exposure - I use a 25 watt incandescent bulb that is 24 inches above the contact frame. I generally expose the film for one second. I will be going to a single point source LED soon.

Chemicals
- Developer – Arista Litho Developer powder (A/B parts)(makes 2 gallons/7.6L – $17.49)(this makes a lot of developer) - Fixer – Arista Arifax Fixer powder (makes 1 gallon/3.8L -$4.99)(this makes a lot of fixer)

Litho film – Arista Litho 2.0 comes in 4X5, 5X7, 8X10 inch sheets, and 25, 50 and 100 sheet boxes.

Equipment needed
- 3 plastic trays (set of 3 - $19.49) - Tongs (set of 3 -$6.99)
- Red safe light bulb ($4.29, the red lights at the hardware store will not work.) All of the above chemicals, film, trays, etc., are available from Freestyle Photographic Supply, Hollywood, Ca. http://www.freestylephoto.biz (Very good company) Cost per session Basically, I get 8 sessions per$18 of developer and $10 of fixer. In other words, the cost per session, for the chemicals, is$3.50.

Preparing the chemicals

Developer – The developer is a two part dry powder. You mix the powder with water and each part makes a gallon of liquid that can be stored for a pretty long time. When you are ready to develop you mix equal parts of "A" and "B" into a tray. A liter fits nicely into an 8 X 10 inch tray.

Once the chemicals are mixed with water, they have a finite shelf life, so I leave them in powder form until I am ready to use them. I take the developer (A/B) and split each part into 8 equal bags (Ziploc baggies). This is about the correct portion to make one liter of developer, (½ L “A” and ½ L “B”). In other words, I will mix one “A” baggy and one “B” baggy into 1 liter of warm tap water. This is normally good for a whole night of developing.

Fixer - I divide the fixer powder into four equal bags. Each bag is about the right amount to make one Liter of fixer. Mix when needed.

Litho film prices:
- 4 X 5, 50 sheets - $10.99 ($0.22 each)
- 5 X 7, 25 Sheets – $9.99 ($0.40 each)
- 8 X 10, 50 sheets - $39.99 ($0.80 each)

So, for less than $10, You can develop 16, 5X7 litho films. Here is a video of how to do it. It would be difficult to see if the only light was the red safe light, so I am going to fake it (lights on) so you can see clearly. When I started using litho film, the quality of my PCBs increased by a tenfold. Last edited: Thread Starter #### Lestraveled Joined May 19, 2014 1,946 Dry Film Resist I mainly use Dupont Riston FX515 dry film resist. It is a negative acting photo resist emulsion sandwiched between two sheets of plastic film. It comes in a roll or sheets. EBay is full of sellers. It costs, depending on the quantity purchased, between 1 and 2 cents per square inch. (That is 35 to 70 cents to sensitize a 5 X 7 inch PCB. (A pre-sensitized 5 X 7 inch positive board from Circuit Specialist costs$8.37.) If I mess up the resist or exposure I am only out 70 cents, not $8.37.) The dry film resist has to be handled in yellow light, meaning no white fluorescents, or bright blue photographic lights. Yellow bug lights, even the CFLs, work just fine. I have three 40 watt CFL yellow bug lights in my shop. The unexposed color is a light blue. Exposed is a dark blue. Below is a picture that shows color verses exposure time of dry film. To expose the film, I use a single 5 Watt, UV, 365nm, LED (driven at 3 watts). Exposure is at 12 inches above the PCB for 9 minutes. A laminator or clothes iron is used to apply the film to the PCB stock. The PCB has to be scrubbed with an abrasive cleaner. I use the cook top cleaner with a scotch bright pad. Cut the film so that it is a little larger than the PCB, an extra ½” seems to work well. This stuff is called “dry film” but it adheres better if you apply it wet. First, you remove one of the plastic protective layers. The official way is to use two pieces of scotch tape, one on each side and pull apart. Or you can do what I do, use a razor knife to abrade one corner until you see a separation, grab the plastic and pull. It comes right off, easy easy. You need a tray of room temperature tap water. Put the clean PCB, copper side up in the tray. Take the film, remove one protective layer as described and put it into the water, resist side down, towards the PCB. Align the PCB under the film and remove from the water. Lightly squeegee the water from between the film and PCB with your fingers. Place on a piece of standard printer paper and feed into the laminator. The PCB should be square with the rollers of the laminator. If you have an infrared thermometer, you want to get the board temperature between 110 and 130 degrees F when it comes out of the laminator (or clothes iron). You can feed it multiple times through the laminator to get the temperature up. Once lamination is complete, take a pair of scissors or a razor knife and cut the excess film and paper off. Be careful not to remove the top protective plastic layer. Also, try not to perturb the top protective layer by cutting too close. Do not remove the top protective layer until after the board is exposed. When the board reaches room temperature it is ready to be exposed. You place the sensitized PCB, resist side up, in the contact printing frame, with the artwork correctly orientated on top of it and then close/place the glass on the top. Position the frame under your UV light source and expose. You need to expose a test strip to determine the correct exposure time. After exposure, the pattern can be inspected. The board is ready to be developed. Remove the top protective film before placing it in the developer. The film is developed in a solution of water and sodium carbonate, 7 - 10 grams (1/2 tablespoon) of sodium carbonate in 1 liter of water. The solution must be warm, 80 to 90 deg. F. (I run closer to 100 deg F.) Use a brush to scrub the board while it is in the developer. The exposed resist is very hard and will not come off. Once developed, rinse the board vigorously under running hot water. Do not develop with cold developer. The most failures that I have had are due to me letting the developer get cold. I build a heated tray just to keep this from happening again. It is amazing how hard the resist gets when it is exposed. Here is a video of the entire dry film process. Sodium carbonate (soda ash) is a few electrons away from sodium bicarbonate, aka baking soda. On EBay you can get 5 pounds of sodium carbonate delivered for about$11, which is enough for over 200, one liter batches.

You can also get solder mask film that handles, exposes and develops just like the dry film resist.

To remove the dry film resist you can use acetone or a solution of water and Sodium Hydroxide (lye/caustic soda/drain cleaner), 40 grams per liter. Sodium Hydroxide is also the developer for most positive resists. Do not use Draino, it is only about 45% sodium hydroxide. 100% Lye (Drain Opener) is available in small hardware stores (Ace). (Check the label)

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#### Lestraveled

Joined May 19, 2014
1,946
Micro Views
The following pictures were taken with a USB microscope. They show some interesting things.

The picture below is of a single laser transparency. Notice the little dot at the end of the pointed trace. At first I though was an anomaly until I took a picture of the 4 layer stack up.

The picture below is of the 4 layer laser transparency stack up. Notice the dot at the end of the pointed trace has grown. Apparently the dot is not an anomaly, it is on more than one of the transparencies. My guess is that it is an artifact from the laser printer. You would expect it to transfer over to the litho film.

Below is the litho film made by the above 4 layer stack up. The dot is not there.

I suspect that the mystery dot does not have enough density to "trigger" the litho film.

I use PCBExpress software to make my PCB artwork. The smallest selectable trace width is 6 mils. The program puts a 3 - 4 mil trace around the pattern to indicate the edge of the PCB. The printed width looks to be more like an 8 mil trace. (The gap between the pad and the fill is 20 mils.) The following pictures show the progression of this trace from the 4 layer stack up to the final PCB.

Resolution

Below is a picture of the 4 layer laser stack up

Below is a picture of the negative litho film made from the above stack up

Below is a picture of the negative resist on the PCB

Below is a picture of the etched PCB.

I would never use a trace this narrow on a home built PCB. It does indicate the resolution of the dry film resist.

Not everything comes out perfectly every time

I didn't see this one until I looked at it with a microscope

The following 2 boards were over exposed on a "bed of tubes" fluorescent exposer. Examples of off contact exposure.

Below are photos of boards made by PCBExpress

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#### Lestraveled

Joined May 19, 2014
1,946
OK, enough about how I make PCBs. Now, lets hear how you do it.

#### jpanhalt

Joined Jan 18, 2008
8,711
First, not all photoresists are the same. Some can be developed with sodium carbonate or sodium silicate, some cannot. The brand I use from DigiKey (Injectorall, http://www.digikey.com/product-search/en/prototyping-products/accessories/2359380?k=INjectoral) cannot be developed with anything less basic than NaOH or KOH. Since the basic chemistry is about the same, i.e., a Novolac resin and a photosentizer, the differences are probably in the photosensitizer and concentrations used.

Second, the best printer in my opinion for the photoresist mask is a good ink jet. It doesn't have to be black. All of the colors block the near UV (UVA) required. Since I now make boards so infrequently, I stopped using the inkjet, because the ink cartridges dried out between uses. I use a laser printer, and rather than layer multiple copies to get adequate density to prevent pin-holing, I just paint over the transparency with a dry erase marker. After that dries, I wipe off the excess with soft tissue. The marker fills the voids and is not removed by the wiping with tissue (Kleenex). Unfortunately, I discovered last year that not all toners will work with that method. HP works great and was the printer I used to develop the method. Brother works poorly. The difference can be seen with a magnifying glass. HP is fine particles. Brother is more like paint splotches.

I develop in dilute caustic soda (NaOH or KOH, about 0.4 molar) and etch with ferric chloride or cupric chloride. I found that the peroxide-based etchants (e.g., H2O2 _HCl) attacked the resist that I use.

I used to use the immersion tin coating, but stopped doing that years ago. Here is an example of a board I made last year with that method:

Here's an example of the dry erase transparency treated and untreated:

Here's a closer view from another project after the dry erase treatment:

And here's the completed board from that transparency:

I don't use thin traces, unless necessary. I estimate the name ("zero width") is about 4 mil or so. I like ferric chloride/cupric chloride because of the sharp edges with minimal undercutting that they give. I have a close up somewhere, but couldn't find it right now. If I do, I will add it to this post.

John

#### Lestraveled

Joined May 19, 2014
1,946
John
Your boards look great! I never thought of using dry erase markers. I will try that.

I agree on using thick traces over thin. No need to make traces thin unless it is really needed.

#### Lestraveled

Joined May 19, 2014
1,946
John
I tried your dry erase marker thing to darken the laser transparency. It darken the image but it also removed laser toner. I tried different ways to apply it but I could not get it to work.

#### jpanhalt

Joined Jan 18, 2008
8,711
I suspect you are not using a LaserJet by HP. My Brother is a great printer, but its fusing temp (?) seems to be lower and the toner is not stable to the dry erase. I got the same sort of results you got. The HP toner is really durable. Since toner doesn't dry out, I am trying to find a place for my HP 4100 MFP. It is/was an expensive and large tank, but works well with the dry erase variation. Brother and HP are the only two manufacturers with which I have experience.

Actually, starting last year, I have pretty much gone to commercial board houses, unless I need something simple in a hurry.

John

#### Lestraveled

Joined May 19, 2014
1,946
John
Yes, I am using an HP 1160. I get my best density with after marker toner cartridges. I put an HP toner cartridge back in and re-tried the dry erase marker. No difference.

I know what you mean about commercial board houses. They have made it so easy, especially PCBExpress. I can still make boards 1/5 to 1/10 the price.

#### jpanhalt

Joined Jan 18, 2008
8,711
Hi Les,

What brand of marker do you use? I will try it with my HP4100 to see if it is the marker or the toner. Deep down, I suspect it is the toner. The newer toners appear to be low temperature versions (energy saving?) of the earlier toners.

As for cost, I just ordered 5 ITEAD boards, 5cmX10cm for \$21.33 total, including shipping. I could not make five boards for that price, including shipping. In fact, one board including shipping would come to about half that. Time and quality will tell.

There will always be times when someone wants a DIY board. So, I think putting down a reasonable procedure for using the photoresist method is a worthwhile project. I have never gotten the toner transfer method to work to my standards.

Regards, John

#### Lestraveled

Joined May 19, 2014
1,946
The dry erase marker I have is a "Messagestor'. It is alcohol based. From the MSDS sheets, most dry erase markers are alcohol based. The "Quartet Duraglide" DE markers appear to be water based. I will try to find some of these.

#### jpanhalt

Joined Jan 18, 2008
8,711
I will try to find the Messagestor brand. The brand I use is Expo from Office Depot. It has a chisel tip rather than pointed. Here's an MSDS for it: https://store.schoolspecialty.com/OA_HTML/xxssi_ibeGetWCCFile.jsp?docName=G616260

Pink has a different MSDS. One thing I should have mentioned is that the black ink seems to behave like a pigment (it becomes a powder when removed); whereas, the others seem to have more dissolved color (i.e., dyes) in them. The colors other than black are more difficult to erase from a white board and do not work in the PCB method either. Expo is sold as "low odor". It is quite possible that the black version has more water in it than the other colors. Water content may not be listed on an MSDS. Anyway, I will try to find the brand you used and test it with my HP printer. That may be a few days, as my next trip into Cleveland won't be until the weekend.

A water-based marker should work better than one that has only alcohol as the carrier.

John

#### Lestraveled

Joined May 19, 2014
1,946
OK, a small update on the dry erase marker issue.

The dry erase markers that contain alcohol or ethanol, take the toner off of the transparencies.
The brands that the above includes are: Expo, Officemax, MessageStor, and Crayloa.

I also tested blue food coloring. It did remove some of the toner. I suspect that it was the Propylene Glycol in it.

I tested tap water. It did not remove any toner at all.

Through the MSDS, I identified that the "Quartet Enduraglide" dry erase markers are alcohol and ethanol free. There are no stores in Tucson that carry them so I ordered some through Amazon. They will arrive Wednesday.

If a simple procedure like applying a dry resist marker to the transparency will greatly improve the density, then the printer to PCB path will get much shorter and simplier. Lets hope.

Come on, lets heard some more ideas.

#### Lestraveled

Joined May 19, 2014
1,946
I received the Quartet Enduraglide water based dry erase markers today and below are the results.

Left image: untouched laser image. Right: Image treated with the Quartet Enduraglide markers.

My evaluation
The water based maker worked better that the alcohol/ethenol based markers. With a very gentle hand, I was able to treat the image without removing any toner, which was a problem that the alcohol markers had. Unfortunately, when removing the marker pigment with a soft cloth or a Q tip, the toner would often come off also.

Conclusion - At first, I was excited about the concept of darkening the image with some kind of ink/pigment. Now, I am not so enthusiastic. I can see that dry erase markers would be good for touch up on small simple boards, in that, if you messed up your touch up, you had a good chance of fixing it. But, I think dry erase markers fail as a wholesale darkening agent for large boards.

Now, how am I going to use all of these markers before they dry out.

#### atferrari

Joined Jan 6, 2004
3,766
WOW, WOW, WOW! Impressive; so much knowledge in play.

Sorry gentlemen. You have experimented a lot. But I am still thinking that I would be happy to get a PCB drilling machine for prototypes. I simply did not buy it (Stepcraft 300) because I run short of time to reach Germany and then catch my flight back home.

After watching locally, a demonstration with a LPKF machine (authorized dealer), I had it in my mind all the time.

/EDIT
it is "PCB milling" machine
EDIT/

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text:
## Precalculus: Mathematics for Calculus, 7th Edition

$\dfrac{x^{2}+2x-3}{x^{2}+8x+16}\cdot\dfrac{3x+12}{x-1}=\dfrac{3x+9}{x+4}$
$\dfrac{x^{2}+2x-3}{x^{2}+8x+16}\cdot\dfrac{3x+12}{x-1}$ Factor both rational expressions completely: $\dfrac{x^{2}+2x-3}{x^{2}+8x+16}\cdot\dfrac{3x+12}{x-1}=\dfrac{(x+3)(x-1)}{(x+4)^{2}}\cdot\dfrac{3(x+4)}{x-1}=...$ Evaluate the product of the two fractions and simplify by removing the factors that appear both in the numerator and in the denominator of the resulting expression: $...=\dfrac{3(x+3)(x-1)(x+4)}{(x+4)^{2}(x-1)}=...$ $...=\dfrac{3(x+3)}{x+4}$ or $\dfrac{3x+9}{x+4}$


text:
HOME  ›   visualization

# Loupe V(D)J Browser: Comparing Multiple Samples

Loupe V(D)J Browser v4.0 and later allows you to load .vloupe files from the cellranger aggr pipeline for multi-sample comparisons.

## Clonotype filtering

For a demo of the different comparisons available, download the two-sample PBMC .vloupe file.

Upon opening the aggregated file in Loupe V(D)J Browser, the views are the same as the ones produced for a single sample file, however, every view can contain data from multiple samples.

Notice that the Filters pop-up window now has a 'Sample' filter.

Clicking on the Select Sample Type drop-down brings up a list of options for dividing the barcodes in the file into samples. These sample name options are based on a column in the aggregation CSV, an input file for the cellranger aggr pipeline. The default options are Donor, Origin, and Library ID, but custom fields like 'Time Points' or 'Disease Status' may also be provided.

For details on the definitions of the default groups look at the Cell Ranger glossary.

To compare the samples in detail, click on the Comparison tab.

### Sample Table

The default view is the Sample Table per donor in the aggregated dataset. The table lists all the loaded V(D)J samples with their Barcode and Clonotype counts. There is also a sparkline of each sample's top 100 clonotype abundances. The table and other charts accessible from the drop-down menu show samples divided by sample groups (donor, origin, library ID, or custom groups).

In this example, there is only one row because the two aggregated libraries come from the same donor:

Above the sample table are all other comparison charts is the control bar. The Sample Group selector can be used to divide the data views in different ways. The tutorial file has the same donor and origin but comes from two different libraries, so toggle to the Library Id option. Using the Chart Type selector, select Top Clonotype Overlap to see the top 10 clonotypes in one library compared to the same clonotypes in the second library. You can change the library whose top 10 Clonotypes you wish to compare using the Top 10 Clonotypes from drop-down menu on the top right.

The top 10 clonotypes from the 10k sample (orange) are also fairly abundant in the 1k sample (blue). You can change the number of clonotypes compared by editing the Num Clonotypes value in the control bar.

### Clonotype Frequency

The Clonotype Frequency chart gives you a feel for how closely the clonotypes of two samples map onto each other. The x-axis represents the frequency of a particular clonotype in sample A, and the y-axis represents the frequency of a particular clonotype in sample B. The size of the circles at each point represents the number of clonotypes that have x barcodes in sample A and y barcodes in sample B.

To get a feel for this, compare the two samples by library ID. The mutual frequencies are lined up vertically in the center of the chart. In this case, the green sample primarily has clonotypes with one barcode, which is where that line falls, but no overlap in clonotypes with multiple barcodes.

You can use the zoom in/out and autoscale buttons on the control bar to zoom into regions of the chart in more detail, and drag the chart around with your mouse.

### Gene Usage comparison

Finally, you can compare the relative frequencies of V, D, J, and C genes across multiple samples by selecting one of those four charts in the chart type selector. For each gene usage chart, you can either look at all chain types or limit to a single chain type through the Chain Type selector.

## Exporting

Most charts have an export button, where the chart image can be exported to SVG or PNG, or the underlying data can be exported to a CSV file (shown in the image above). When the file has a large number of samples, another selector will appear that can be used to select a subset of samples to view throughout the charts.

## Next steps

This concludes the Loupe V(D)J Browser tutorial. We hope that you are comfortable enough now to use this tool on your Chromium Single Cell 5′ V(D)J data. If you have matching 5′ Gene Expression data, the Loupe V(D)J Browser and Loupe Browser are more powerful together. See the integrated V(D)J and gene expression analysis tutorial.

If you encounter difficulties or have general questions about the software, email [email protected] with your questions.


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# Mew

6,222pages on
this wiki

Mew
(ミュウ Myuu)
Artwork General
National Dex: #151
Evolves From: None
Evolves Into: None
Generation: Generation I
Pronunciation: $ME-UU$
Icon(s):
Base Stats Biological Details
HP: 100 Species: New Species Pokémon
Attack: 100 Type(s): Psychic
Defense: 100 Height: 1'04"
Special Atk: 100 Weight: 8.8 lbs.
Special Def: 100 Abilities: Synchronize
Speed: 100 Pokédex Color: Pink
Stat Total: 600 Gender: Genderless
Footprint:
Cry:

Mew (Japanese: ミュウ Myuu) is a Psychic-type Mythical and Legendary Pokémon, and is the last Pokémon in the Kanto Pokédex. Unlike any other Pokémon, it has the ability to learn every TM and HM except Draco Meteor.

Mew does not evolve into or from any other Pokémon; however, its DNA was cloned to create Mewtwo. It was the last Pokémon in the Pokédex until Generation II was released.

## Appearance

Mew's appearance is that of a pink cat-like Pokémon with large feet, blue eyes, small pointed ears and a long, skinny tail. It is covered with a layer of fine pink hair, only visible with the aid of a microscope. It has short arms with three fingers on each paw.

## Special abilities

Mew has the Ability Synchronize. This ability allows it to pass on any status problem, except Sleep and Frozen, to its opponent. This ability also raises the encounter rate of Pokémon with its same nature.

Additionally, it is said that Mew's DNA possesses the genetic composition of all existing Pokémon species, thus allowing it to use all known Pokémon techniques. Also, as shown in Lucario and the Mystery of Mew, it can change into any Pokémon using Transform. It changed into Meowth, Ho-Oh, Pidgeot, Pikachu, Aipom, Treecko, Taillow, Mime Jr., and Pichu.

Mew has also shown its ability to Levitate, create force fields, and change its form into any existing species of Pokémon. It is a very versatile and powerful Pokémon.

## Appearances

### In the anime

Mew has been featured in several Pokémon movies, having a large role in most of them. Mew first appeared in the movie Mewtwo Strikes Back or Pokémon the First Movie, and fought with Mewtwo.

Mew's second movie appearance was in Lucario and the Mystery of Mew, where a different Mew from the first movie takes Meowth and Ash's Pikachu to the Tree of Beginning. Mew has also made appearances in other movies. Mew has been shown several times though in the TV series anime. Mew is in the opening theme of the Pokémon anime series, and it was featured in the TV Movie special Mewtwo Returns, during a flashback sequence of Mewtwo's. In the episode A Lean, Mean, Team Rocket Machine, it appears in one of Meowth's boss fantasies, along with Celebi, Phione, Manaphy, and Jirachi. It appears as a mirage Pokémon in the special The Mastermind of Mirage Pokémon.

### In the manga

The first legendary Pokémon that Red saw was Mew; he tried to capture it, but due to his lack of experience he was unable to catch it. In the Emerald arc, Emerald used soil from Mew's birthplace to calm down all the rental Pokémon as Mew is believed to have the DNA of all Pokémon.

## Game info

### Game locations

Version(s) Location Rarity Mew Locations Red/Blue Mew Glitch One Yellow Mew Glitch One Gold/Silver Time Capsule None Crystal Time Capsule None Ruby/ Sapphire Trade None Emerald Faraway Island One FireRed/ LeafGreen Trade None Diamond/ Pearl My Pokémon Ranch One Platinum My Pokémon Ranch One HeartGold/SoulSilver Mystery Gift, Trade None Black/White Trade None

### Pokédex entries

Pokédex Entries
Red
Blue
So rare that it is still said to be a mirage by many experts. Only a few people have seen it worldwide.
Yellow
When viewed through a microscope, this Pokémon's short, fine, delicate hair can be seen.
Gold
Apparently, it appears only to those people who are pure of heart and have a strong desire to see it.
Silver
Its DNA is said to contain the genetic codes of all Pokémon, so it can use all kinds of techniques.
Crystal
Because it can learn any move, some people began research to see if it is the ancestor of all Pokémon.
Ruby
Mew is said to possess the genetic composition of all Pokémon. It is capable of making itself invisible at will, so it entirely avoids notice even if it approaches people.
Sapphire
Mew is said to possess the genetic composition of all Pokémon. It is capable of making itself invisible at will, so it entirely avoids notice even if it approaches people.
Emerald
A Mew is said to possess the genes of all Pokémon. It is capable of making itself invisible at will, so it entirely avoids notice even if it approaches people.
FireRed
A Pokémon of South America that was thought to have been extinct. It is very intelligent and learns any move.
LeafGreen
So rare that it is still said to be a mirage by many experts. Only a few people have seen it worldwide.
Diamond
Because it can use all kinds of moves, many scientists believe Mew to be the ancestor of Pokémon.
Pearl
Because it can use all kinds of moves, many scientists believe Mew to be the ancestor of Pokémon.
Platinum
Because it can use all kinds of moves, many scientists believe Mew to be the ancestor of Pokémon.
HeartGold
Apparently, it appears only to those people who are pure of heart and have a strong desire to see it.
SoulSilver
Its DNA is said to contain the genetic codes of all Pokémon, so it can use all kinds of techniques.
Black
Because it is able to use every move, there are many scientists who believe that it is the ancestor of all Pokémon.
White
Because it is able to use every move, there are many scientists who believe that it is the ancestor of all Pokémon.
Black 2
Because it is able to use every move, there are many scientists who believe that it is the ancestor of all Pokémon.
White 2
Because it is able to use every move, there are many scientists who believe that it is the ancestor of all Pokémon.

### Super Smash Bros.

Mew has been a featured Pokémon in every one of the Super Smash Brothers games.

In original Super Smash Bros., a Mew will fly up without affecting the battlefield. In Super Smash Bros. Melee there is a 1 in 251 chance that a Poké Ball item contains Mew, and if this is the first time the player sees it, they will get an "achievement" after the battle. While it does not attack, the player receives 10,000 points after the battle is over. In Super Smash Bros. Brawl, it may even leave the player with a CD, a rare item that unlocks additional music in the game. Also, like in Melee, if it is the player's first time seeing Mew, they will get a "Notice" record from encountering him. This can be viewed later, along with the date and time the player received it.

## Sprites

Version
Red Blue Yellow Red (JP) Green (JP) Back
Gold Silver Crystal Back
Ruby Sapphire Emerald FireRed LeafGreen Back
Diamond Pearl Platinum HeartGold SoulSilver Back
Black White Back

## Origins

Mew's name likely has to do with the onomatopoeia for a kitten's cry, referring to its cat-like appearance. However, some say that it is derived from the greek letter μ(pronounced mew). In some fonts they way it is written makes it look like a cat.

## Gallery

 Mew (anime)Mew's original artworkMew seen in the Pokémon Theme Song (Gotta Catch 'Em All).Add a photo to this gallery

## Trivia

• Mew was the very first Pokémon trademark ever to be applied for and registered, which may be the reason why it was made "the ancestor of all Pokémon."
• Although it can learn all moves, Mew can't learn powerful and rare moves such as Blast Burn, Judgment, and Freeze Shock, mainly because some moves are to be learned by a specific Pokémon, and Mew cannot learn them under a specific condition.
• Mew's gender has been a subject of debate for some time. Officially, Mew is neither; however, in various media such as the films and television series, Mew is described as mother. That it gave birth to Mewtwo (as described in the video game series) isn't in a literal sense. Mewtwo was created from Mew's DNA; thus, Mew could be considered Mewtwo's "parent." In PokéPark Wii: Pikachu's Adventure, it is referred to as a "he".
• It is one of only two Pokémon able to learn Transform, the other being Ditto. This also supports the fact that Mew is the ancestor of all Pokemon.
• Based on scales, Mew has the longest tail in relation to its own body. This does not include snakelike Pokémon such as Ekans, Arbok, and Seviper.
• Although having lower stats than Mewtwo, in the anime it is shown to be just as powerful, if not more powerful, than its clone counterpart. This is another example of Anime physics.
• In Japan, a pink Nintendo DS was released with Mew on it.
• In Pokémon Red, Blue and Yellow you cannot complete the PokéDex without doing the Mew Glitch.
• Mew was the first Pokémon introduced who is not legitimately obtainable in the main series. Celebi soon followed.

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text:
## Course Outline

• segmentGetting Started (Don't Skip This Part)
• segmentStatistics and Data Science: A Modeling Approach
• segmentPART I: EXPLORING VARIATION
• segmentChapter 1 - Welcome to Statistics: A Modeling Approach
• segmentChapter 2 - Understanding Data
• segmentChapter 3 - Examining Distributions
• segmentChapter 4 - Explaining Variation
• segmentPART II: MODELING VARIATION
• segmentChapter 5 - A Simple Model
• segmentChapter 6 - Quantifying Error
• segmentChapter 7 - Adding an Explanatory Variable to the Model
• segmentChapter 8 - Digging Deeper into Group Models
• segmentChapter 9 - Models with a Quantitative Explanatory Variable
• segmentPART III: EVALUATING MODELS
• segmentChapter 10 - The Logic of Inference
• segmentChapter 11 - Model Comparison with F
• segmentChapter 12 - Parameter Estimation and Confidence Intervals
• segmentFinishing Up (Don't Skip This Part!)
• segmentResources

### list High School / Advanced Statistics and Data Science I (ABC)

Book
• High School / Advanced Statistics and Data Science I (ABC)
• High School / Statistics and Data Science I (AB)
• High School / Statistics and Data Science II (XCD)
• College / Statistics and Data Science (ABC)
• College / Advanced Statistics and Data Science (ABCD)
• College / Accelerated Statistics and Data Science (XCDCOLLEGE)
• Skew the Script: Jupyter

## 6.9 Using the Normal Model to Make Predictions

Let’s go back now to the question we asked earlier: Given the distribution of thumb length, what is the probability that the next randomly sampled student would have a thumb length at least 65.1 mm long? (The “at least” phrase indicates a thumb length 65.1 mm or even longer!) How could we use the smoothed out normal model to answer this question?

### Fitting the Normal Model to a Distribution of Data

We first need to fit the normal model to the distribution of thumb length in our data (much like how we fit simple shapes over the irregularly shaped state of California). Below is the jagged distribution of Thumb. We have represented the empty model (the mean of the distribution, 60.1) as a blue line to serve as a reference point.

empty_model <- lm(Thumb ~ NULL, data = Fingers)

gf_dhistogram(~ Thumb, data = Fingers) %>%
gf_model(empty_model)

The normal distribution is actually a family of theoretical distributions, each having a different mean and a different standard deviation. To find the particular normal distribution that best fits our data, we need to find the two parameters (mean and standard deviation) that define the best-fitting curve, i.e. the curve that minimizes the squared residuals of our data from the model.

The figure below can give you a sense of how normal distributions can vary from each other, depending on their means and standard deviations. Note that three of the four distributions pictured have the same mean, which is 0, but quite different shapes. The fourth distribution has a mean that is below the other three.

Finding the normal curve that best fits our distribution of data is fairly straightforward. First we use our sample mean and standard deviation ($$\bar{Y}$$ and $$s$$) as estimates of the population mean and standard deviation ($$\mu$$ and $$\sigma$$). Then, using these estimates we can use the gf_fitdistr() function to graphically overlay the best-fitting normal model on the distribution of data.

The gf_fitdistr() function can overlay a number of different mathematical probability distributions. The default distribution is a normal distribution (this function automatically calculates the mean and standard deviation to draw the normal curve).

gf_dhistogram(~ Thumb, data = Fingers) %>%
gf_model(empty_model) %>%
gf_fitdistr()

You might be thinking—hmm, the normal curve doesn’t fit that well over the data. You are correct! But remember, our goal is not just to model our data, but to model the long-term results of the DGP. Even when our data look jagged, the population distribution may not be.

And, the goal of a model is not to fit perfectly, but to balance the fit of the model with the simplicity and elegance of the model. By modeling error with the normal curve, we can trade in the complexity of 157 jagged values for an elegant, two-parameter model—the normal curve.

Finally, remember our original question? We wanted to know the exact probability that the next randomly sampled student would have thumb length of more than 65.1 mm. To help visualize this question, let’s add one more thing to the graph: a red line to locate 65.1 mm in reference to the distribution.

gf_dhistogram(~ Thumb, data = Fingers) %>%
gf_model(empty_model) %>%
gf_fitdistr() %>%
gf_vline(xintercept = 65.1, color = "red")

With the help of this picture, we can restate our question as: What proportion of the area under the blue normal curve falls to the right of the red vertical line? This proportion, which we will calculate below, would tell us the probability that the next randomly sampled student would have a thumb length longer than 65.1 mm.

### Using the Normal Distribution to Calculate Probabilities

Once you have fit a smooth normal curve over a data distribution, you can use mathematical properties of the curve (don’t worry, we’ll let R handle that!) to find the probability of scores falling into specific regions. This is exactly like what we did when we used rectangles to model the area of California. Once we fit a rectangle, we could simply use the formula for the area of a rectangle ($$l \times w$$) to calculate the area.

In the graph below, we have removed the distribution of data, and left only the best-fitting normal curve. The answer to the question, “What is the probability that the next student will have a thumb length more than 65.1 mm?,” is represented by the region colored green under the curve. If we can find the proportion of total area that is colored green, we can use that proportion as our probability estimate for the next observation being greater than 65.1.

Because the normal distribution has a smooth shape easily defined by two parameters, R can tell us the exact probability by using the mathematical function for a normal distribution. There is a function called xpnorm() that can do this with just three pieces of information: the border you are interested in, and the mean and standard deviation of the normal distribution.

xpnorm(65.1, mean(Fingers$Thumb), sd(Fingers$Thumb))

Note that you can also include argument labels, mean = and sd =, which would give you the flexibility to put the values in a different order like this: xpnorm(65.1, sd = sd(Fingers$Thumb), mean = mean(Fingers$Thumb)).

Try running the function in the code window below. It will generate both this picture as well as some output where you can read off the exact probability that you are looking for.

require(coursekata) # try running this code xpnorm(65.1, mean(Fingers$Thumb), sd(Fingers$Thumb)) xpnorm(65.1, mean(Fingers$Thumb), sd(Fingers$Thumb))ex() %>% check_error()
CK Code: B2_Code_UsingNormal_01


text:
# 3.2 Asymptotics

## Practice with Big O

Find the least nonnegative integer, $$n$$, such that $$f(x$$) is $$O(x^{n})$$ when $$f$$ is defined by each of the expressions below.

If there is none, enter "none".

1. $$f(x) = 2x^{3} + x^{2}\log x$$
Exercise 1
$$2x^{3}$$ grows as fast as $$x^{3}$$, and $$x^2\log x$$ grows strictly slower than $$x^{3}$$.

2. $$f(x) =2x^{2} + x^{3}\log x$$
Exercise 2
$$2x^{2}$$ grows as fast as $$x^{2}$$, and $$x^{3}\log x$$ grows strictly faster than $$x^{3}$$ but strictly slower than $$x^{4}$$.

3. $$f(x) =(1.1)^{x}$$
Exercise 3
$$(1.1)^{x}$$ grows strictly faster than any polynomial.

4. $$f(x) = (0.1)^{x}$$
Exercise 4
As $$x$$ goes to infinity, $$(0.1)^{x}$$ goes to 0. So it grows strictly slower than any constant (same as a polynomial of degree 0).

5. $$f(x) = \dfrac{x^{4} + x^{2} + 1}{x^{3} + 1}$$
Exercise 5
This fraction grows as fast as $$x^{4}/x^{3}=x$$.

6. $$f(x) = \dfrac{x^{4} + 5 \log x}{x^{4} + 1}$$
Exercise 6
This fraction grows as fast as $$x^{4}/x^{4}=1$$.

7. $$f(x) = 2^{3 \log_{2}x^{2}}$$
Exercise 7
$$2 ^{3 \log_{2}x^{2}} = 2^{\log_{2} (x^{2})^{3}} = 2^{\log_{2}x^{2 \cdot 3}} = 2^{\log_{2}x^{6}} = x^{6}$$