Covariance and the Correlation Coefficient

Prove that

Prove that independence of the random variables $\xi$ and $\eta$ implies $\operatorname {Cov}\left[ \xi , \eta \right] = 0$.

Prove that:

Construct an example of two dependent random variables whose correlation coefficient is zero.

Find $\operatorname {Cov}\left[ \xi , \eta \right]$ for the random variables with the distribution table given in problem:

Let the random variable $\xi$ take the values 0, 1, and $-1$ with probabilities $1/3$, $1/2$, and $1/6$ respectively. Find the correlation coefficient between $\xi$ and $\xi^2$.

Let the random variables $\xi$ and $\eta$ be independent, with $\mathbb {P}\left\{ \xi =1\right\} = \mathbb {P}\left\{ \xi =-1\right\} = 1/2$, $\mathbb {P}\left\{ \eta =1\right\} = \mathbb {P}\left\{ \eta =-1\right\} = 1/4$, and $\mathbb {P}\left\{ \eta =0\right\} = 1/2$. Are the random variables $\xi \eta$ and $\eta$:

Find the correlation coefficient between the number of «ones» rolled and the number of «sixes» rolled in $n$ rolls of a fair die.

A random point $(\xi , \eta )$ has a uniform distribution in the square $\left\{ (x, y) : \left|x\right| + \left|y\right| \leqslant \sqrt{2}\right\}$.

A random point $(\xi , \eta )$ has a uniform distribution in the triangle with vertices at the points $(0, 0)$, $(2, 1)$, $(2, 0)$.

A random point $(\xi , \eta )$ is uniformly distributed inside the unit disk centered at the origin.

The joint distribution of the random variables $\xi$ and $\eta$ has density

Two points are thrown at random onto the segment $[0, a]$. Find:

Find the correlation coefficient between the random variables $\xi$ and $\eta = e^{-\xi }$, if $\xi$ has the standard normal distribution.

The random variables $\xi$ and $\eta$ are independent and normally distributed with parameters $a$ and $\sigma^2$. Find:

Let $\xi$ be a random variable with a symmetric distribution and finite variance. Find the correlation coefficient of the random variables $\xi$ and $\left|\xi \right|$.

Let the random variables $\xi_1, \ldots , \xi_{n+m}$ be independent, identically distributed, and have finite variance. For $n > m$, find the correlation coefficient between the sums $\xi_1 + \ldots + \xi_n$ and $\xi_{m+1} + \ldots + \xi_{m+n}$.

Give an example of two random variables $\xi$ and $\eta$ such that $\mathbb {E}\left[\xi \right]$ and $\mathbb {E}\left[\eta \right]$ exist, but $\mathbb {E}\left[\xi \eta \right]$ does not exist.

An urn contains $a$ white and $b$ black balls. $k$ balls are drawn from the urn simultaneously, $k \leqslant a+b$. Find the expectation and variance of the number of white balls drawn.

For a group of $n$ people, find the expectation and variance of the number of days in the year on which no one's birthday falls.

$n$ letters are written, addressed to different recipients. There are $n$ envelopes with the corresponding addresses. The letters are placed into the envelopes in a random order. Find the mean value and variance of the number of letters sent to the correct address.

Prove that there do not exist three random variables $\xi$, $\eta$, and $\zeta$ such that the correlation coefficient of any two of them equals $-1$.

Let $\xi_1, \ldots , \xi_n$ be random variables such that the correlation coefficient of any two of them equals the same number $\rho$. Prove that $\rho \geqslant -1/(n-1)$.

Let the random variables $\xi$ and $\eta$ have zero mean values, unit variances, and correlation coefficient $\rho$. Show that $\mathbb {E}\left[\max (\xi^2, \eta^2)\right] \leqslant 1 + \sqrt{1-\rho^2}$.

Let the random variables $\xi$ and $\eta$ have zero mean values, unit variances, and correlation coefficient $\rho$, $0 < \rho < 1$. Find all values of $a$ and $b$ for which the random variables $\xi - a\eta$ and $\eta - b\xi$ are uncorrelated.