Expectation (Other Cases), Covariance

Give an example of two dependent random variables $\xi , \eta$ whose covariance equals 0, such that $\xi , \eta$ are Gaussian.

Let $\xi , \eta$ be random variables with nonzero variances. Do there exist nonzero numbers $a, b$ such that the random variable $a \xi +b \eta$ is independent of both $\xi$ and $\eta$?

A random variable $\xi$ has the following distribution function:

Compute the expectation and variance of $\xi$.

A random variable $\xi$ has the following distribution function:

A random variable $\eta \sim \operatorname {Bin}(1,0.5)$ (Bernoulli distributed with success probability 0.5) is defined so that $\xi -\eta$ and $\eta$ are independent. Compute $\mathbb {E}\left[2^{\xi +\eta }\right]$.

Random variables $\xi , \eta$ (possibly dependent) have finite variances: $\operatorname {Var}\left[\xi \right]=\sigma_{1}^{2}, \operatorname {Var}\left[\eta \right]=\sigma_{2}^{2}$. Specify the range within which $\operatorname {Var}\left[\xi +\eta \right]$ can vary.

A shooter at a shooting range fires into a quarter circle, i.e. into the region $D=\left\{ (x, y): x^{2}+y^{2}<1, x>0, y>0\right\}$. The random vector $(\xi , \eta )^T$ is the shooter's point of impact and is uniformly distributed on $D$. Find $\operatorname {cov}(\xi , \eta )$.

A source generates two signals at random moments of time. Let the first signal be generated at time $X$, and the second — after an additional time $Y$. The random variables $X$ and $Y$ are independent and have exponential distribution with parameter $\lambda$ (i.e. their densities equal $\lambda e^{-\lambda t} \; \mathbb {1}_{t \geq 0}$). Let $U=X+Y$ be the moment at which the second signal is emitted, and $V=X/(X+Y)$ be the fraction of the time that was needed to generate the first signal. Is it true that this fraction $V$ is independent of the total time $U$? Find the distributions of the random variables $U$ and $V$, as well as $\operatorname {Cov}\left[ U, V \right]$.

A random vector $(\xi , \eta )$ has density

Compute the covariance matrix of the random vector $(\xi , \eta )$. What is the distribution of the random variable $\xi$?