text: k x k =0. Vector area of parallelogram = a vector x b vector This gives us Since i, j, k are unit vectors of fixed length we can use the result from the previous section and write As a result, This formula reduces to the formula given in the previous section if A is of fixed magnitude (length), since dA x /dt, dA y /dt, dA z /dt all equal zero. Vectores en el plano • Los vectores i → = (1, 0) y j → = (0, 1) son vectores unitarios que tienen, respectivamente, la dirección del eje X y el eje Y, y sentido positivo. Coefficients of i, j ,k are added seperately,and the resultant value will also be a vector. The magnitude of a vector can be found using Pythagoras's theorem. The vector , being the sum of the vectors and , is therefore This formula, which expresses in terms of i, j, k, x, y and z, is called the Cartesian representation of the vector in three dimensions. This engineering statics tutorial goes over how to use the i, j, k unit vectors to express any other vector. The i, j, and k fields are multiplied together and then all values are added up to give the total dot product. p = 3i + j, q = -5i + j. Find the area of the parallelogram whose two adjacent sides are determined by the vectors i vector + 2j vector + 3k vector and 3i vector − 2j vector + k vector. Since the vectors are given in i, j form, we can easily calculate the resultant. b vector = 3i vector − 2j vector + k vector. Using $i,j,$ and $k$ for the standard unit vectors goes back to Hamilton (1805–1865) and his invention of quaternions $\mathbf H$ in the 1840s. Example. Now, take the vector derivative of A with respect to time. As curl or rotation of two vectors give the direction of third vector. If using this calculator for a 3D vector, then the user enters in all fields. The formula Find p + q. The Magnitude of a Vector. This could also have been worked out from a diagram: The Magnitude of a Vector. 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text:
# The frog problem with negative steps

### Standard Problem description

In this question The Frog Problem (puzzle in YouTube video) a frog has to jump from leaf to leaf on a row of leaves. And the question is how long it takes on average to reach the end.

In that specific case the frog only jumps to the leaves in front of him with each leaf having equal probability. It is computed that the expectation value for the number of steps to reach the end is $$J_n = \sum_{k=1}^n 1/k$$, when the frog has $$n$$ leaves in front of him.

### New extended problem

But what is the solution when the frog can also remain still and go one step backwards. (there are infinitely many leaves behind the frog, the game only ends when the frog has no leaves in front of him)

This would lead to a recurrence relation like: $$J_{n} = (n+1) J_{n-1} – n J_{n-2}-1$$

To make the solution final we need to know $$J_0$$ and $$J_1$$.

It is trivial that the expected number of steps for a frog with zero leaves in front of him is 0 ($$J_0 = 0$$).

But what is $$J_1$$? What is the expected number of steps for a frog that has only one leaf to go?

### Derivation/intuition of the recurrence relation:

$$J_{n+1} = (n+2) J_{n} – (n+1) J_{n-1}-1$$

There are $$n+2$$ leaves to go to. The $$n$$ leaves in front of the frog and the one leaf on which the frog is sitting is the same situation as the frog who has $$n-1$$ leaves in front of him/her. The one leaf backwards results in the frog being in the situation of a frog that has $$n+1$$ leaves in front of him/her but he took an extra step.

$$J_{n} = \frac{n+1}{n+2} {J_{n-1}} + \frac{1}{n+2} {(J_{n+1}+1)}$$

### Solution attempt 1

It seems like the solution is close to $$J_n = c + \sum_{k=1}^n 1/k$$ with some constant $$c$$… but not exactly. When I fill that expression into the recurrence relation then I get to:

$$\begin{array}{rcl} J_{n} &=& (n+1) J_{n-1} – n J_{n-2}-1 \\ c + \sum_{k=1}^n \frac{1}{k} &=& (n+1) \left(c + \sum_{k=1}^{n-1} \frac{1}{k} \right) – n \left(c + \sum_{k=1}^{n-2} \frac{1}{k} \right) -1 \\ \sum_{k=1}^n \frac{1}{k} &=& (n+1) \left(\sum_{k=1}^{n-1} \frac{1}{k} \right) – n \left(\sum_{k=1}^{n-2} \frac{1}{k} \right) -1 \\ \frac{1}{n} + \frac{1}{n-1} &=& (n+1)\frac{1}{n -1} -1\\ \frac{1}{n} + \frac{1}{n-1} &=& \frac{2}{n -1} \\ \end{array}$$

### Solution attempt 2

Simulation by Markov chain (this results into something that looks like $$J_n = c + \sum_{k=1}^n 1/k$$ but as shown before that can not be true.)

nm <- 50
library(expm)

for (n in 1:40) {
# stochastic Matrix
M <- pracma::Toeplitz(c(rep(1,nm+1)),c(1,1,rep(0,nm-1))) / c(2:(nm+2))
M[1,1:2] <- c(1,0)

# positions of frogs after k steps
V <- c(rep(0,n),1,rep(0,nm-n))
Vm <- sapply(0:nn, FUN = function(k) V %*% (M %^% k))

# mean number of steps by computing 1-F(0)
E <- sum(1-Vm[1,])
ev[n] <- E
}

n <- 1:40
plot(n,ev,xlim=c(0,20))

title("simulated \n expected number of steps",cex.main=1)

H <- cumsum(1/n)
mod <- lm(ev-H~1)
lines(n,H+coef(mod)[1])

coef(mod)

legend(0,6.6, c("simulation","2.325547 + H_n") ,cex=0.7, lty=c(NA,1), pch = c(1,NA))

Jn <- ev[-c(1,2)]
Jn1 <- ev[-c(1,40)]
Jn2 <- ev[-c(39,40)]
np <- n[-c(1,2)]

plot(Jn, (np+1)*Jn1 - (np) * Jn2 -1,
xlab = expression(J[n]),
ylab = expression((n+1)%*%J[n-1]-n%*%J[n-2]-1 ))
lines(c(0,10),c(0,10))

title("testing recurrence relation", cex.main=1)

In this answer to the simpler solution. The motion of the frog is not computed by using the recurrence relation, but instead by writing down the probability distribution where the frog might be after $$k$$ jumps.

In that case the distribution is like a diffusing wave, which will eventually be absorbed completely in the final leaf. In this case we can not compute it because there is a tiny number of frogs that will never reach the goal. But maybe we solve the puzzle with this starting point by finding some explicit solution or by changing the expression to include the backwards leaves?

### Solution for $$J_1$$

But what is J1? What is the expected number of steps for a frog that has only one leaf to go?

The solution is $$J_1 = 2(e-1)$$ and other terms $$J_n$$ can be expressed as a sum.

### Rewriting the recurrence relation as a sum

The recurrence relation is not gonna solve the problem entirely (because one term in the initial conditions is not known), but it does allow us to express $$J_n$$ as an expression in terms of a finite sum.

The recurrence relation can be rewritten. (for n>3)

$$J_n – J_{n-1} = n (J_{n-1} – J_{n-2})-1$$

let $$D_n = J_n – J_{n-1}$$

$$D_n = n D_{n-1}-1$$

and with starting point $$D_2 = 2x$$ and we can write (note the recurrence relation is a bit different for $$n = 2$$ as @quester noted in the comments):

$$\begin{array}{rcrcrcrcrcr} D_1 &=& 3 + 2\,x \\\\ D_2 &=& 2\,x\\ D_3 &=& \overbrace{6 \,x}^{\times 3} &-&1\\ D_4 &=& \rlap{\overbrace{\hphantom{\begin{array}5040\,x&-&7\cdot 6\cdot 5\cdot 4 \end{array}}}^{\times 4}} 24\,x&-&4 &-& 1 \\ D_5&=& \rlap{\overbrace{\hphantom{\begin{array}5040\,x&-&7\cdot 6\cdot 5\cdot 4 &-& 7\cdot 6\cdot 5 \end{array}}}^{ \times 5}} 120\,x&-&5\cdot 4 &-& 5 &-& 1 \\\\ D_6&=& 720\,x&-&6\cdot 5\cdot 4 &-& 6\cdot 5 &-& 6 &- & 1 \\\\ D_7&=& 5040\,x&-&7\cdot 6\cdot 5\cdot 4 &-& 7\cdot 6\cdot 5 &-& 7\cdot 6 &- & 7 &-&- 1 \\\\ D_k &=& k! x &-&\rlap{\sum_{l=3}^{k} \frac{k!}{l!}} \\ \end{array}$$

and

$$J_n = x \sum_{k=1}^n k! -\sum_{k=3}^n\sum_{l=3}^{k} \frac{k!}{l!}$$

### Closed form expression for $$x$$

Lets rewrite $$D_k$$

$$\begin{array}{} D_k &=& k! x – \sum_{l=3}^{k} \frac{k!}{l!}\\ &=& k! \left(x – \sum_{l=0}^k \frac{1!}{l!} – 2.5 \right)\end{array}$$

If we conjecture that $$\lim_{k \to \infty }D_k$$ is positive and finite then this leads to the requirement $$\lim_{k \to \infty }\left(x – \sum_{l=0}^k \frac{1!}{l!} – 2.5 \right)= 0$$ and

$$x = \lim_{k \to \infty } \left(\sum_{l=0}^k \frac{1!}{l!} – 2.5\right) = e-2.5 \approx 0.2182818$$

The argument that $$\lim_{k \to \infty }D_k$$ is finite is still a conjecture but it seems plausible to me.

### Closed form expression for $$D_k$$

Filling in $$x$$ into the expression of $$D_k$$ will lead to:

$$\begin{array}{} J_1 &=& D_1 & = & 2e-2 \\ &&D_2 & = & 2e-5 \\ &&D_k & = & k! \left( e – \sum_{l=0}^k \frac{1}{l!}\right) \\ \end{array}$$

### Is $$J_n$$ finite?

We can argue that $$J_n$$ (the mean number of steps to reach the finish) is finite for any starting point $$n$$, because the mean position from the finish is decreasing to zero bounded by an exponential decay.

• The mean distance from the finish: Say a frog starts in position $$x$$. Then after one jump the frog will be somewhere in position $$0 \leq y \leq x+1$$ (each option with probability $$\frac{1}{x+2}$$), and if $$y \neq 0$$ then after after two jumps the frog will be somewhere in position $$0 \leq z \leq y+1$$ (each option with probability $$\frac{1}{y+2}$$). Then the mean position $$\bar{z}$$ of a frog that started at $$x$$ and makes two jumps will be:
$$\sum_{y=1}^{x+1}\sum_{z=1}^{y+1}\frac{z}{(x+2)(y+2)} = \frac{x^2+5x+4}{4x+8} \underbrace{\leq\frac{10}{12}x}_{\text{if x\geq1}}$$
So whatever the position of the frog, after two jumps, he will be on average at least 1/6 closer to the finish.

• Probability a frog is still in the game: Note that the probability of a frog still in the game relates to the mean distance of a frog in the game. The mean distance after $$k$$ jumps is $$\mu_{\bar{x},k} \sum_{x=1}^\infty x f(x,k)$$, where $$f(x,k)$$ is the probability of the frog to be in position $$x$$ after $$k$$ jumps. The probability of a frog to be still in the game is: $$1-F_{(\text{finished at or before jump k})}(k)=\sum_{x=1}^\infty f(x,k) < \mu_{\bar{x},k} \leq x_{start} \left(\frac{10}{12} \right)^{k/2}$$.

• Finiteness of $$J_n$$ The mean number of steps necessary can be found by $$\sum_{k=0}^\infty k f(k)$$ with $$f(k)$$ the probability that it takes $$k$$ steps. But you can also take $$\sum_{k=0}^\infty 1-F(x)$$ with $$F(k)$$ the probability that it takes $$k$$ or less steps (note that the integral of the CDF is related to the mean or more generaly the expected value of any quantity is related to the quantile function). And since $$1−F(k)$$ is smaller than some decreasing exponential function of $$k$$, so must be the sum smaller than the integral/sum of that function and that is finite.

### Starting with a simpler problem

With the recurrence relation $$D_n = n D_{n-1} – 1$$ it is problematic to solve the case because the starting condition is not defined.

We can instead try to pose a simpler problem (suggested in the comments by @quester and @Hans). Let’s say that there are only $$m+2$$ leaves (instead of infinite), and thus the frog with only $$m$$ leaves in front of him will not be able to jump backwards. Then $$J_m = J_{m-1}$$ (the frog in point $$m$$ has the same options as the frog in point $$m-1$$) and we will have

$$D_{m} = m! \left(x_m – \sum_{l=0}^m \frac{1!}{l!} – 2.5 \right) = 0$$

which gives a solution for $$x_{m}$$ as:

$$x_m = \sum_{l=0}^m \frac{1!}{l!} – 2.5$$

and the limit of $$x_m$$ as we start adding leaves is:

$$\lim_{m\to \infty} x_m = \lim_{m\to \infty} \sum_{l=0}^m \frac{1!}{l!} -2.5 = e-2.5$$

format: plain


text:
# Joint equation of pair of lines joining the origin and the points of intersection of a line and curve

My teacher told me that if I am given a line $$L_1$$ and a curve $$C_1$$ $$L_1 : lx + my + n=0$$ $$C_1 : ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$$

If $$n \not= 0$$ and if the line $$L_1$$ cuts the curve $$C_1$$ at $$2$$ points, then the joint equation of straight lines through these points and passing the origin is given by "homogenizing" the equation of the curve by the equation of the line. Which would give:

$$ax^2 + 2hxy + by^2 + \left(2gx+2fy\right)\left(\frac{lx+my}{-n}\right) + c\left(\frac{lx+my}{-n}\right)^2 = 0$$

She didn't go into much detail about this but told us that it has something to do with making making the degree of every term in the equation the same or making it "homogeneous".

Why does this work and why does it give me the joint equation of the $$2$$ required straight lines? Are there any sources through which I can do some further reading on this topic?

You can rewrite the homogenized equation as $$Ax^2+Bxy+Cy^2=0$$ and it can be factorized as $$(p_1x+q_1y)(p_2x+q_2y)=0$$. So it represents the line pair $$p_1x+q_1y=0$$ and $$p_2x+q_2y=0$$.
Since the points of intersection satisfy both $$C_1$$ and $$L_1$$, they satisfy the homogenized equation.
• But why does it pass through the origin and the two points in which the line $L_1$ and the curve $C_1$ intersect? – eem Jul 7 '19 at 2:44
• $(0,0)$ satisfies both $p_1x+q_1y=0$ and $p_2x+q_2y=0$. So it passes through the origin. Take a point of intersection $P(x_0,y_0)$ of $C_1$ and $L_1$. We have $lx_0 + my_0 + n=0$ and $ax_0^2 + 2hx_0y_0 + by_0^2 + 2gx_0 + 2fy_0 + c = 0$. So we have $ax_0^2 + 2hx_0y_0 + by_0^2 + \left(2gx_0+2fy_0\right)\left(\frac{lx_0+my_0}{-n}\right) + c\left(\frac{lx_0+my_0}{-n}\right)^2 = 0$. $P$ satisfies the homogenized equation. – CY Aries Jul 7 '19 at 2:49


text:
## University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson

# Chapter 9 - Section 9.4 - Comparison Tests - Exercises - Page 509: 52

Converges

#### Work Step by Step

Consider $a_n=\dfrac{\sqrt[n] n}{n^2}$ and $b_n=\dfrac{1}{ n^2}$ Now, $\lim\limits_{n \to \infty}\dfrac{a_n}{b_n} =\lim\limits_{n \to \infty}\dfrac{\dfrac{\sqrt[n] n}{n^2}}{\dfrac{1}{ n^2}}$ or, $=\lim\limits_{n \to \infty} \sqrt[n] n$ and $\lim\limits_{n \to \infty} n^{1/n}=1 \ne 0 \ne \infty$ Here, $\Sigma_{n=1}^\infty \dfrac{1}{n^2}$ is a convergent p-series with $p \gt 1$ Thus, the series converges by the limit comparison test.

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
# Table of content for Dictionary

I have one problem in the latex \tableofcontents.

Look at the image I circled.

For Dictionary from A-Z               1-250

How to do that in latex in \tableofcontents?

Is it possible in LaTeX?

-

Well, the command \addtocontents{toc}{STUFF} adds anything you want to the table of contents. You need to format your entry yourself, however.

If you look into TOC of a LaTeX file, you see that its entries have the following structure \contentsline {SECTION}{\numberline {NUMBER}TITLE}{PAGE}{HYPERLINK NAME} (the last argument is there only if you use hyperref). So you need to:

1. Put \label{first_page} on the first page.

2. Put \label{last_page} on the last page

3. Put on the first page \addtocontents{toc}{\contentsline {section}{\numberline{} Words}{\pageref{first_page}--\pageref{last_page}}{}}

You many need to change section to chapter here depending on the style you want.

Good luck!

-

Your information are really spare. The modification based also on the packages which creates the index. For example you can use makeidx, scrindex or splitidx ...

In my example I uses the class scrbook in combination with scrindex:

\documentclass[ngerman,idxtotoc]{scrbook}
\usepackage{babel}
\usepackage{scrindex}
\usepackage{etoolbox}
\usepackage[user,abspage]{zref}
\usepackage{hyperref}
\makeatletter
\if@openright\cleardoublepage\else\clearpage\fi%
\@mkboth{\indexname}{\indexname}%
}
\phantomsection
{\usekomafont{chapterentry}%
\protect\hfill%
\zpageref{startindex} -- \zpageref{endindex}%
}}
}
\AtEndEnvironment{theindex}{\zlabel{endindex}}
\makeindex
\begin{document}
\tableofcontents
\chapter{foo}
Text\index{Text}
\printindex
Test
\end{document}

-


text:
# Question #30549

Jan 30, 2018

Frequency

#### Explanation:

The pitch of sound depends on the frequency of the wave.

Frequency = number of oscillations( or the cycles) per second.

More the frequency, higher is the pitch of the sound.

Less frequency corresponds to low pitch.


text:
# Correlation does not imply causation

A chart that, according to Bobby Henderson, correlates the number of pirates with global temperature. The two variables are correlated, but one does not imply the other.

Correlation does not imply causation is a phrase in science and statistics that emphasizes that a correlation between two variables does not necessarily imply that one causes the other.[1][2] Many statistical tests calculate correlation between variables. A few go further and calculate the likelihood of a true causal relationship; examples are the Granger causality test and convergent cross mapping.

The counter assumption, that correlation proves causation, is considered a questionable cause logical fallacy in that two events occurring together are taken to have a cause-and-effect relationship. This fallacy is also known as cum hoc ergo propter hoc, Latin for "with this, therefore because of this", and "false cause". A similar fallacy, that an event that follows another was necessarily a consequence of the first event, is sometimes described as post hoc ergo propter hoc (Latin for "after this, therefore because of this").

In a widely studied example, numerous epidemiological studies showed that women who were taking combined hormone replacement therapy (HRT) also had a lower-than-average incidence of coronary heart disease (CHD), leading doctors to propose that HRT was protective against CHD. But randomized controlled trials showed that HRT caused a small but statistically significant increase in risk of CHD. Re-analysis of the data from the epidemiological studies showed that women undertaking HRT were more likely to be from higher socio-economic groups (ABC1), with better-than-average diet and exercise regimens. The use of HRT and decreased incidence of coronary heart disease were coincident effects of a common cause (i.e. the benefits associated with a higher socioeconomic status), rather than cause and effect, as had been supposed.[3]

As with any logical fallacy, identifying that the reasoning behind an argument is flawed does not imply that the resulting conclusion is false. In the instance above, if the trials had found that hormone replacement therapy caused a decrease in coronary heart disease, but not to the degree suggested by the epidemiological studies, the assumption of causality would have been correct, although the logic behind the assumption would still have been flawed.

## Usage

In logic, the technical use of the word "implies" means "to be a sufficient circumstance". This is the meaning intended by statisticians when they say causation is not certain. Indeed, p implies q has the technical meaning of the material conditional: if p then q symbolized as p → q. That is "if circumstance p is true, then q follows." In this sense, it is always correct to say "Correlation does not imply causation."

However, in casual use, the word "imply" loosely means suggests rather than requires. The idea that correlation and causation are connected is certainly true; where there is causation, there is a likely correlation. Indeed, correlation is used when inferring causation; the important point is that such inferences are made after correlations are confirmed as real and all causational relationship are systematically explored using large enough data sets.

Edward Tufte, in a criticism of the brevity of "correlation does not imply causation", deprecates the use of "is" to relate correlation and causation (as in "Correlation is not causation"), citing its inaccuracy as incomplete.[1] While it is not the case that correlation is causation, simply stating their nonequivalence omits information about their relationship. Tufte suggests that the shortest true statement that can be made about causality and correlation is one of the following:[4]

• "Empirically observed covariation is a necessary but not sufficient condition for causality."
• "Correlation is not causation but it sure is a hint."

## General pattern

For any two correlated events, A and B, the following relationships are possible:

• A causes B;
• B causes A;
• A and B are consequences of a common cause, but do not cause each other;
• There is no connection between A and B; the correlation is coincidental.

Less clear-cut correlations are also possible. For example, causality is not necessarily one-way; in a predator-prey relationship, predator numbers affect prey, but prey numbers, i.e. food supply, also affect predators.

The cum hoc ergo propter hoc logical fallacy can be expressed as follows:

1. A occurs in correlation with B.
2. Therefore, A causes B.

In this type of logical fallacy, one makes a premature conclusion about causality after observing only a correlation between two or more factors. Generally, if one factor (A) is observed to only be correlated with another factor (B), it is sometimes taken for granted that A is causing B, even when no evidence supports it. This is a logical fallacy because there are at least five possibilities:

1. A may be the cause of B.
2. B may be the cause of A.
3. some unknown third factor C may actually be the cause of both A and B.
4. there may be a combination of the above three relationships. For example, B may be the cause of A at the same time as A is the cause of B (contradicting that the only relationship between A and B is that A causes B). This describes a self-reinforcing system.
5. the "relationship" is a coincidence or so complex or indirect that it is more effectively called a coincidence (i.e. two events occurring at the same time that have no direct relationship to each other besides the fact that they are occurring at the same time). A larger sample size helps to reduce the chance of a coincidence, unless there is a systematic error in the experiment.

In other words, there can be no conclusion made regarding the existence or the direction of a cause-and-effect relationship only from the fact that A and B are correlated. Determining whether there is an actual cause-and-effect relationship requires further investigation, even when the relationship between A and B is statistically significant, a large effect size is observed, or a large part of the variance is explained.

## Examples of illogically inferring causation from correlation

### B causes A (reverse causation)

Example 1
The faster windmills are observed to rotate, the more wind is observed to be.
Therefore wind is caused by the rotation of windmills. (Or, simply put: windmills, as their name indicates, are machines used to produce wind.)

In this example, the correlation (simultaneity) between windmill activity and wind velocity does not imply that wind is caused by windmills. It is rather the other way around, as suggested by the fact that wind doesn’t need windmills to exist, while windmills need wind to rotate. Wind can be observed in places where there are no windmills or non-rotating windmills—and there are good reasons to believe that wind existed before the invention of windmills.

### A causes B and B causes A (bidirectional causation)

Increased pressure is associated with increased temperature.
Therefore pressure causes temperature.

The ideal gas law, $PV=nRT$, describes the direct relationship between pressure and temperature (along with other factors) to show that there is a direct correlation between the two properties. For a fixed volume and mass of gas, an increase in temperature causes an increase in pressure; likewise, increased pressure causes an increase in temperature. This demonstrates bidirectional causation. The conclusion that pressure causes temperature is true but is not logically guaranteed by the premise.

### Third factor C (the common-causal variable) causes both A and B

Main article: Spurious relationship

All these examples deal with a lurking variable, which is simply a hidden third variable that affects both causes of the correlation; for example, the fact that it is summer in Example 3. A difficulty often also arises where the third factor, though fundamentally different from A and B, is so closely related to A and/or B as to be confused with them or very difficult to scientifically disentangle from them (see Example 4).

Example 1
Sleeping with one's shoes on is strongly correlated with waking up with a headache.
Therefore, sleeping with one's shoes on causes headache.

The above example commits the correlation-implies-causation fallacy, as it prematurely concludes that sleeping with one's shoes on causes headache. A more plausible explanation is that both are caused by a third factor, in this case going to bed drunk, which thereby gives rise to a correlation. So the conclusion is false.

Example 2
Young children who sleep with the light on are much more likely to develop myopia in later life.
Therefore, sleeping with the light on causes myopia.

This is a scientific example that resulted from a study at the University of Pennsylvania Medical Center. Published in the May 13, 1999 issue of Nature,[5] the study received much coverage at the time in the popular press.[6] However, a later study at Ohio State University did not find that infants sleeping with the light on caused the development of myopia. It did find a strong link between parental myopia and the development of child myopia, also noting that myopic parents were more likely to leave a light on in their children's bedroom.[7][8][9][10] In this case, the cause of both conditions is parental myopia, and the above-stated conclusion is false.

Example 3
Illustration of example 3: a correlation between ice cream consumption and crime, but shows that the actual cause is temperature.
As ice cream sales increase, the rate of drowning deaths increases sharply.
Therefore, ice cream consumption causes drowning.

The aforementioned example fails to recognize the importance of time and temperature in relationship to ice cream sales. Ice cream is sold during the hot summer months at a much greater rate than during colder times, and it is during these hot summer months that people are more likely to engage in activities involving water, such as swimming. The increased drowning deaths are simply caused by more exposure to water-based activities, not ice cream. The stated conclusion is false.

Example 4
A hypothetical study shows a relationship between test anxiety scores and shyness scores, with a statistical r value (strength of correlation) of +.59.[11]
Therefore, it may be simply concluded that shyness, in some part, causally influences test anxiety.

However, as encountered in many psychological studies, another variable, a "self-consciousness score", is discovered that has a sharper correlation (+.73) with shyness. This suggests a possible "third variable" problem, however, when three such closely related measures are found, it further suggests that each may have bidirectional tendencies (see "bidirectional variable", above), being a cluster of correlated values each influencing one another to some extent. Therefore, the simple conclusion above may be false.

Example 5
Since the 1950s, both the atmospheric CO2 level and obesity levels have increased sharply.
Hence, atmospheric CO2 causes obesity.

Richer populations tend to eat more food and consume more energy

Example 6
HDL ("good") cholesterol is negatively correlated with incidence of heart attack.
Therefore, taking medication to raise HDL decreases the chance of having a heart attack.

Further research[12] has called this conclusion into question. Instead, it may be that other underlying factors, like genes, diet and exercise, affect both HDL levels and the likelihood of having a heart attack; it is possible that medicines may affect the directly measurable factor, HDL levels, without affecting the chance of heart attack.

## Determining causation

Main articles: Causality and Causality (physics)

The point of view that correlation implies causation may be regarded as a theory of causality, which is somewhat inherent to the field of statistics. Within academia as a whole, the nature of causality is systematically investigated from several academic disciplines, including philosophy and physics.

In academia, there is a significant number of theories on causality; The Oxford Handbook of Causation (Beebee et al. 2009) encompasses 770 pages. Among the more influential theories within philosophy are Aristotle's Four causes and Al-Ghazali's occasionalism.[13] David Hume argued that causality is based on experience, and experience similarly based on the assumption that the future models the past, which in turn can only be based on experience – leading to circular logic. In conclusion, he asserted that causality is not based on actual reasoning: only correlation can actually be perceived.[14] Immanuel Kant, according to Beebee et al., held that "a causal principle according to which every event has a cause, or follows according to a causal law, cannot be established through induction as a purely empirical claim, since it would then lack strict universality, or necessity".[15]

Outside the field of philosophy, theories of causation can be identified in classical mechanics, statistical mechanics, quantum mechanics, spacetime theories, biology, social sciences, and law.[15] To establish a correlation as causal within physics, it is normally understood that the cause and the effect must connect through a local mechanism (cf. for instance the concept of impact) or a nonlocal mechanism (cf. the concept of field), in accordance with known laws of nature.

From the point of view of thermodynamics, universal properties of causes as compared to effects have been identified through the Second law of thermodynamics, confirming the ancient, medieval and Cartesian[16] view that "the cause is greater than the effect" for the particular case of thermodynamic free energy. This, in turn, is challenged by popular interpretations of the concepts of nonlinear systems and the butterfly effect, in which small events cause large effects due to, respectively, unpredictability and an unlikely triggering of large amounts of potential energy.

### Causality construed from counterfactual states

Intuitively, causation seems to require not just a correlation, but a counterfactual dependence. Suppose that a student performed poorly on a test and guesses that the cause was his not studying. To prove this, one thinks of the counterfactual – the same student writing the same test under the same circumstances but having studied the night before. If one could rewind history, and change only one small thing (making the student study for the exam), then causation could be observed (by comparing version 1 to version 2). Because one cannot rewind history and replay events after making small controlled changes, causation can only be inferred, never exactly known. This is referred to as the Fundamental Problem of Causal Inference – it is impossible to directly observe causal effects.[17]

A major goal of scientific experiments and statistical methods is to approximate as best possible the counterfactual state of the world.[18] For example, one could run an experiment on identical twins who were known to consistently get the same grades on their tests. One twin is sent to study for six hours while the other is sent to the amusement park. If their test scores suddenly diverged by a large degree, this would be strong evidence that studying (or going to the amusement park) had a causal effect on test scores. In this case, correlation between studying and test scores would almost certainly imply causation.

Well-designed experimental studies replace equality of individuals as in the previous example by equality of groups. The objective is to construct two groups that are similar except for the treatment that the groups receive. This is achieved by selecting subjects from a single population and randomly assigning them to two or more groups. The likelihood of the groups behaving similarly to one another (on average) rises with the number of subjects in each group. If the groups are essentially equivalent except for the treatment they receive, and a difference in the outcome for the groups is observed, then this constitutes evidence that the treatment is responsible for the outcome, or in other words the treatment causes the observed effect. However, an observed effect could also be caused "by chance", for example as a result of random perturbations in the population. Statistical tests exist to quantify the likelihood of erroneously concluding that an observed difference exists when in fact it does not (for example see P-value).

### Causality predicted by an extrapolation of trends

When experimental studies are impossible and only pre-existing data are available, as is usually the case for example in economics, regression analysis can be used. Factors other than the potential causative variable of interest are controlled for by including them as regressors in addition to the regressor representing the variable of interest. False inferences of causation due to reverse causation (or wrong estimates of the magnitude of causation due the presence of bidirectional causation) can be avoided by using explanators (regressors) that are necessarily exogenous, such as physical explanators like rainfall amount (as a determinant of, say, futures prices), lagged variables whose values were determined before the dependent variable's value was determined, instrumental variables for the explanators (chosen based on their known exogeneity), etc. See Causality#Statistics and Economics. Spurious correlation due to mutual influence from a third, common, causative variable, is harder to avoid: the model must be specified such that there is a theoretical reason to believe that no such underlying causative variable has been omitted from the model. In particular, underlying time trends of both the dependent variable and the independent (potentially causative) variable must be controlled for by including time as another independent variable.[citation needed]

## Use of correlation as scientific evidence

Much of scientific evidence is based upon a correlation of variables[19] – they are observed to occur together. Scientists are careful to point out that correlation does not necessarily mean causation. The assumption that A causes B simply because A correlates with B is often not accepted as a legitimate form of argument.

However, sometimes people commit the opposite fallacy – dismissing correlation entirely, as if it does not suggest causation at all. This would dismiss a large swath of important scientific evidence.[19] Since it may be difficult or ethically impossible to run controlled double-blind studies, correlational evidence from several different angles may be the strongest causal evidence available.[20] The combination of limited available methodologies with the dismissing correlation fallacy has on occasion been used to counter a scientific finding. For example, the tobacco industry has historically relied on a dismissal of correlational evidence to reject a link between tobacco and lung cancer.[21]

Correlation is a valuable type of scientific evidence in fields such as medicine, psychology, and sociology. But first correlations must be confirmed as real, and then every possible causative relationship must be systematically explored. In the end correlation can be used as powerful evidence for a cause-and-effect relationship between a treatment and benefit, a risk factor and a disease, or a social or economic factor and various outcomes. But it is also one of the most abused types of evidence, because it is easy and even tempting to come to premature conclusions based upon the preliminary appearance of a correlation.

Correlations are used in Bell's theorem to disprove local causality.

## References

1. ^ a b
2. ^ Aldrich, John (1995). "Correlations Genuine and Spurious in Pearson and Yule". Statistical Science 10 (4): 364–376. doi:10.1214/ss/1177009870. JSTOR 2246135.
3. ^ Lawlor DA, Davey Smith G, Ebrahim S (June 2004). "Commentary: the hormone replacement-coronary heart disease conundrum: is this the death of observational epidemiology?". Int J Epidemiol 33 (3): 464–7. doi:10.1093/ije/dyh124. PMID 15166201.
4. ^ Tufte, Edward R. (2003). The Cognitive Style of PowerPoint. Cheshire, Connecticut: Graphics Press. p. 4. ISBN 0-9613921-5-0.
5. ^ Quinn GE, Shin CH, Maguire MG, Stone RA (May 1999). "Myopia and ambient lighting at night". Nature 399 (6732): 113–4. doi:10.1038/20094. PMID 10335839.
6. ^ CNN, May 13, 1999. Night-light may lead to nearsightedness
7. ^ Ohio State University Research News, March 9, 2000. Night lights don't lead to nearsightedness, study suggests
8. ^ Zadnik K, Jones LA, Irvin BC, et al. (March 2000). "Myopia and ambient night-time lighting". Nature 404 (6774): 143–4. doi:10.1038/35004661. PMID 10724157.
9. ^ Gwiazda J, Ong E, Held R, Thorn F (March 2000). "Myopia and ambient night-time lighting". Nature 404 (6774): 144. doi:10.1038/35004663. PMID 10724158.
10. ^ Stone, J; et al., E; Held, R; Thorn, F (March 2000). "Myopia and ambient night-time lighting". Nature 404 (6774): 144. doi:10.1038/35004665.
11. ^ The Psychology of Personality: Viewpoints, Research, and Applications. Carducci, Bernard J. 2nd Edition. Wiley-Blackwell: UK, 2009.
12. ^ Ornish, Dean. "Cholesterol: The good, the bad, and the truth" [1] (retrieved 3 June 2011)
13. ^ Beebee et al., 2009
14. ^ David Hume (Stanford Encyclopedia of Philosophy)
15. ^ a b Beebee et al. 2009
16. ^ Lloyd, A.C., The principle that the cause is greater than its effect, Pronesis 21(2), 1976
17. ^ Holland, Paul W. (1986). "Statistics and Causal Inference". Journal of the American Statistical Association 81 (396): 945–960.
18. ^ Pearl, Judea (2000). Causality: Models, Reasoning, and Inference. Cambridge University Press.
19. ^ a b Novella. "Evidence in Medicine: Correlation and Causation". Science and Medicine. Science-Based Medicine.
20. ^ http://www.michaelnielsen.org/ddi/if-correlation-doesnt-imply-causation-then-what-does/
21. ^ http://www.sciencebasedmedicine.org/evidence-in-medicine-correlation-and-causation/


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# Match Making Groups of Players

Given a situation where I have a pool of X people of different scores (ranks) S with game size N. How do I partition the pool into N sized games where the standard deviation of ranks for each games is minimized?

(quality of the game is inversely proportional to STD of ranks)

• Possible duplicate of gamedev.stackexchange.com/questions/464/… though a smaller subset. – The Communist Duck Dec 22 '10 at 15:47
• @The Communist Duck I don't really want a framework. I'm looking to code it myself, so I need an algorithm (or heuristics). – Kendall Hopkins Dec 22 '10 at 16:05
• It's not clear what you want to happen when N|X fails. Does N vary over some range? Can I make games with anywhere between 2 and N players? Are you looking to minimize the mean, total, or some other function of the standard deviations? – user744 Dec 23 '10 at 11:10
• @Joe Wreschnig You can assume N % X == 0. Any N is an acceptable number, but N = 1 is the trivial case. – Kendall Hopkins Dec 23 '10 at 15:33

If your values for X and N are very large, it would probably make the most sence to keep the player list sorted by Rank such that you could always take the bottom N elements to create a group. This will front-load your computation time (keeping the list sorted every time you add a new item), in contrast with the above algorithm which would require execution every time a new group is to be created.
• I've looked over this algorithm a bit and I can't understand the logic or it's result at all. Could you explain it a bit more? Why would I want a groupMembers filled with the index of players with the smallest rank? I've reworded the question, maybe that'll help make what I'm asking for more clear. – Kendall Hopkins Dec 23 '10 at 0:57


text:
# Solving Ramanujan’s Puzzling Problem

Consider a sequence of functions as follows:-

$f_1 (x) = \sqrt {1+\sqrt {x} }$
$f_2 (x) = \sqrt{1+ \sqrt {1+2 \sqrt {x} } }$

$f_3 (x) = \sqrt {1+ \sqrt {1+2 \sqrt {1+3 \sqrt {x} } } }$

……and so on to

$f_n (x) = \sqrt {1+\sqrt{1+2 \sqrt {1+3 \sqrt {\ldots \sqrt {1+n \sqrt {x} } } } } }$

Evaluate this function as n tends to infinity.

Or logically:

Find

$\displaystyle{\lim_{n \to \infty}} f_n (x)$ .

### Solution

Ramanujan discovered

$$x+n+a=\sqrt{ax + (n+a)^2 +x \sqrt{a(x+n)+(n+a)^2 +(x+n) \sqrt{\ldots}}}$$

which gives the special cases

$$x+1=\sqrt{1+x \sqrt{1 + (x+1) \sqrt{1 + (x+2) \sqrt{1 + (x+2) \sqrt{\ldots}}}}}$$

for x=2 , n=1 and a=0

$$3= \sqrt{1+2 \sqrt{1+3 \sqrt{1+ 4 \sqrt{1+\cdots}}}}$$

Comparing these two expressions & assuming

=$X$ , we can write the problem as:

$\displaystyle {\lim_{n \to \infty}} f_n (x)$

= $\sqrt {1+X}$

= $\sqrt {1+3}$

=$\sqrt {4}$

=$2$

For further info please refer the comments below. There is also a supportive article on Ramanujan Nested Radicals on this blog.

1. thank you , this is one of my try out question in my school.
quiet confused since i saw this crazy square root .lol

2. thank you , this is one of my try out question in my school.
quiet confused since i saw this crazy square root .lol

3. second equation is wrong x+1=sq(1+xsq(1+(x+1))sq(1+(x+2)sq(1+(x+3))))

4. second equation is wrong x+1=sq(1+xsq(1+(x+1))sq(1+(x+2)sq(1+(x+3))))

5. Ramanujan always the best

6. Ramanujan always the best

7. I think you might be one of the best bloggers in India today. We are having a TEDx conference, and it would be great to have you as a Speaker. I am sure you can come up with a very interesting talk. Let me know however I can contact you.

8. I think you might be one of the best bloggers in India today. We are having a TEDx conference, and it would be great to have you as a Speaker. I am sure you can come up with a very interesting talk. Let me know however I can contact you.

9. wow – wouldnt have a clue where to start!

10. wow – wouldnt have a clue where to start!

11. Hi there,

Just a small typo — I think you meant to write the limit as n tends to infinity. On all of the limits you wrote in that article, you unfortunately said that x goes to infinity.

x_x

• x_x Corrected Now. Thanks.

12. Hi there,

Just a small typo — I think you meant to write the limit as n tends to infinity. On all of the limits you wrote in that article, you unfortunately said that x goes to infinity.

x_x

• x_x Corrected Now. Thanks.

13. There are two slightly different versions of this nested radical, so you need to be careful.

The version posed by Ramanujan was
sqrt(1 + 2*sqrt(1 + 3*sqrt(1 + 4*sqrt(1 + … = 3

Your version is almost the same:
sqrt(1 + 1*sqrt(1 + 2*sqrt(1 + 3*sqrt(1 + … = sqrt(1 + 3) = 2.

14. There are two slightly different versions of this nested radical, so you need to be careful.

The version posed by Ramanujan was
sqrt(1 + 2*sqrt(1 + 3*sqrt(1 + 4*sqrt(1 + … = 3

Your version is almost the same:
sqrt(1 + 1*sqrt(1 + 2*sqrt(1 + 3*sqrt(1 + … = sqrt(1 + 3) = 2.

15. Answer is 3 according to formulation 27 at http://mathworld.wolfram.com/NestedRadical.html .

16. Answer is 3 according to formulation 27 at http://mathworld.wolfram.com/NestedRadical.html .

17. You’re right..! After using google, I got this Link , which was also saying the same. But I wasn’t satisfied.

18. You’re right..! After using google, I got this Link , which was also saying the same. But I wasn’t satisfied.

19. I’m not entirely sure, but using C++ (with n = 1,000,000) I numerically evaluated it to the function f(x) = 2. But as I said, not quite sure!

20. I’m not entirely sure, but using C++ (with n = 1,000,000) I numerically evaluated it to the function f(x) = 2. But as I said, not quite sure!

Get up to 80% discount on various products with exclusive coupons


text:
SERVING THE QUANTITATIVE FINANCE COMMUNITY

SWilson
Topic Author
Posts: 5
Joined: February 13th, 2018, 5:27 pm

### Conditional Strike Option

Came across am option that is effectively a collar, where the call/ceiling option let's say is $1.5, straight up. The put/floor option lets say is$1.25, fine.  However, if the underlying goes below the put option strike ($1.25), the actual intrinsic settlement uses a 3rd rate called the "conditional" strike which lets say is$1.35 (a value between the floor and ceiling) NOT the \$1.25.  Trying to price this is a meaningful but simple way.  It does not have a knockout, barrier, corridor or other exotic type feature.  Any thoughts on breaking this down to use a standard-type valuation for the time value?

Alan
Posts: 9234
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Re: Conditional Strike Option

By "standard-type valuation", I will first assume you mean under a GBM process. All you need is the joint density of (ST,MT):  the terminal value and the minimum of the process over (0,T).

That density is known in closed form for drifting BM and hence GBM. (Googling shows it's in Mark Joshi's 'Concepts.." book, for example)

Then, the option value is simply the (discounted, double) integral of that density times the payoff function that you gave in your post. (Do the integral numerically and you're done -- if a GBM value is all you really wanted).

Of course, maybe you'll want to go on from there to a more market-consistent model. It would be nice to have the stand-alone ST density agree with the market's RN density which you can infer from Breeden-Litzenberger, for example. Can you infer a model-independent, market-consistent joint (ST,MT) density?  Hard to say --  likely depends on what other types of options are liquid and well-quoted. You may need to calibrate various more complicated models to the vanilla (and other liquid barrier) options, and then do Monte Carlos to price the thing.

travisf
Posts: 14
Joined: May 12th, 2015, 8:51 pm

### Re: Conditional Strike Option

It sounds like this is a two leg option: a 1.5 strike call together with a 1.35 strike put with a knockin barrier at 1.25.

Alan
Posts: 9234
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Re: Conditional Strike Option

Good observation -- simpler than mine and a way to use standard formulas (for GBM).

SWilson
Topic Author
Posts: 5
Joined: February 13th, 2018, 5:27 pm

### Re: Conditional Strike Option

Alan, this is helpful.  Yes, using GBM.  So can you elaborate on how I would adjust variables as it relates to N(d1) and N(d2) that you mentioned using a joint density, when we have a put option that has the 2 prices above?  To start I'm going to take some type of average and then go from there.  Also assuming S(t) will be 1.25 since the moneyness is based on that price.  But in reality, intrinsically, the strike is 1.35.

SWilson
Topic Author
Posts: 5
Joined: February 13th, 2018, 5:27 pm

### Re: Conditional Strike Option

Yes, or like travisf says, it operates like a simultaneous knock-in/knock-out.  I was thinking 2 options with a 1.35 strike with a knock-in AND knock-out at 1.25 .

Alan
Posts: 9234
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Re: Conditional Strike Option

No, I believe travisf is correct. For the put(s) it is entirely (under GBM) a knock-in with a knock-in barrier at 1.25 and strike of 1.35. Under GBM, which is a continuous process (and assuming continuous monitoring), it's impossible for the separate strike of 1.25 to be effective without triggering the knock-in event.

So, just go look up standard formulas for that one.
Last edited by Alan on February 14th, 2018, 5:19 pm

SWilson
Topic Author
Posts: 5
Joined: February 13th, 2018, 5:27 pm

THANKS!


text:
# ZEU-EXP/AT

 2017-04-2709:33 [PUBDB-2017-01945] Journal Article Klepser, S. The optimal on-source region size for detections with counting-type telescopes Astroparticle physics 89, 10 - 13 (2017) [10.1016/j.astropartphys.2017.01.005]   Source detection in counting type experiments such as Cherenkov telescopes often involves the application of the classical Eq. (17) from the paper of Li & Ma (1983) to discrete on- and off-source regions [...] Restricted: PDF; 2017-04-1016:48 [PUBDB-2017-01615] Journal Article et al Gamma-ray observations under bright moonlight with VERITAS Astroparticle physics 91, 34 - 43 (2017) [10.1016/j.astropartphys.2017.03.001]   Imaging atmospheric Cherenkov telescopes (IACTs) are equipped with sensitive photomultiplier tube (PMT) cameras. Exposure to high levels of background illumination degrades the efficiency of and potentially destroys these photo-detectors over time, so IACTs cannot be operated in the same configuration in the presence of bright moonlight as under dark skies [...] Restricted: PDF; 2017-03-1316:12 [PUBDB-2017-01367] Journal Article et al Observations of M31 and M33 with the $Fermi$ Large Area Telescope: A Galactic Center Excess in Andromeda? The astrophysical journal 836(2), 208 (2017) [10.3847/1538-4357/aa5c3d]   The Fermi Large Area Telescope (LAT) has opened the way for comparative studies of cosmic rays (CRs) and high-energy objects in the Milky Way (MW) and in other, external, star-forming galaxies. 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text:
2019
Том 71
№ 11

# A Problem with Condition Containing an Integral Term for a Parabolic-Hyperbolic Equation

Abstract

In a layer obtained as the Cartesian product of an interval $[−T_1 ,T_2], T_1 ,T_2 > 0$, and a space $ℝ_p, p ≥ 1$, we study a problem with nonlocal condition in the time variable containing an integral term for a mixed parabolic-hyperbolic equation in the class of functions almost periodic in the space variables. For this problem, we establish a criterion of uniqueness and sufficient conditions for the existence of solutions. To solve the problem of small denominators encountered in the construction of the solution, we use the metric approach.

English version (Springer): Ukrainian Mathematical Journal 67 (2015), no. 5, pp 723-734.

Citation Example: Kuz A. M., Ptashnik B. I. A Problem with Condition Containing an Integral Term for a Parabolic-Hyperbolic Equation // Ukr. Mat. Zh. - 2015. - 67, № 5. - pp. 635-644.

Full text


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# Z-score and Hypothesis Testing [closed]

I have a large dataset of vehicles's speeds. My goal is to infer an upper bound on the (average) speed of the population of all vehicles $$U$$

The way this is to be done is like this:
Given a potential speed bound $$S$$ we will test the hypothesis that $$U < S$$. The null hypothesis then would be: $$U >= S$$

I've a rather limited statistics knowledge (I'm a programmer!) and would like to know an algorithm for accepting or rejecting an $$S$$.

We have no knowledge of the population of all vehicles (neither distribution nor mean).

I know I have to use z-score (it's a large dataset), but I'm having trouble figuring out what each variable in the z-score formula should correspond to in my case. I'm thinking maybe I should look at only the vehicles where $$\text{speed of vehicle} >= S$$ for testing the null hypothesis, this would be the sample set, and the population would be all vehicles.

• How would taxi speeds be representative of bus speeds? – Glen_b Dec 31 '18 at 2:28
• Apart from the concerns about representativity, what is really the null hypothesis? Do you have postulated an upper limit? How? The legal upper speed limit, or something else? – kjetil b halvorsen Dec 31 '18 at 11:55
• We are concerned because we know these "irrelevant" constraints are crucial to obtaining a correct answer. You're headed in an erroneous direction where no amount of theory, calculations, or programming are going to do you any good. – whuber Dec 31 '18 at 18:48
• If you are really just interested in the upper bound on speed (and not just average speed), why not just google the fastest car in the world (among the population of all cars)? You don't even need statistics here if the answer is already well-known. It seems, for an appropraitely defined universe, U is given by 284.6 mph by the Koenigsegg Agera RS:foxnews.com/auto/… – StatsStudent Dec 31 '18 at 19:07
• The upper bound on speed is c, the speed of light. If you find an maximum speed for a population of traveling vehicles, that is nice, but the meaning is a bit empty. Just taking the maximum of a population will do that but one typically asks for some percentage of the population that exceeds a given speed as this is more likely to be a representative number, i.e., more independent of population size. For example, 5% exceed 127 kph, or some such. I think you need to explore the goal, aim, purpose, rational usage, or form of the question before positing it. Currently, the question is unclear. – Carl Dec 31 '18 at 19:32