Dependence

Let

Show that $f$ is a continuous joint density function, but that the (first) marginal density function $g(x) = \int_{-\infty }^{\infty } f(x, y) d y$ is not continuous.

Let $Q = \left\{ q_{n}: n \geq 1\right\}$ be a set of real numbers, and define

Show that $f_{Q}$ is a continuous joint density function whose first marginal density function is discontinuous at the points in $Q$. Can you construct a continuous joint density function whose first marginal density function is continuous nowhere?

Two grids of parallel lines are superimposed: the first grid contains lines distance $a$ apart, and the second contains lines distance $b$ apart which are perpendicular to those of the first set. A needle of length $r( < \min \left\{ a, b\right\} )$ is dropped at random. Show that the probability it intersects a line equals $r(2 a+2 b-r) /(\pi a b)$.

The plane is ruled by the lines $y = n$, for $n = 0, \pm 1, \ldots$, and onto this plane we drop a cross formed by welding together two unit needles perpendicularly at their midpoints.

Let $X$ and $Y$ be independent random variables each having the uniform distribution on $[0,1]$. Let $U = \min \left\{ X, Y\right\}$ and $V = \max \left\{ X, Y\right\}$. Find $\mathbb {E}\left[U\right]$, and hence calculate $\operatorname {Cov}\left[U, V\right]$.

Three points A, B, C are chosen independently at random on the circumference of a circle. Let $b(x)$ be the probability that at least one of the angles of the triangle ABC exceeds $x \pi$. Show that

Hence find the density and expectation of the largest angle in the triangle.

Let $\left\{ X_{r}: 1 \leq r \leq n\right\}$ be independent and identically distributed with finite variance, and define $\bar{X} = n^{-1} \sum_{r = 1}^{n} X_{r}$. Show that $\operatorname {Cov}\left[\bar{X}, X_{r}-\bar{X}\right] = 0$.

Let $X$ and $Y$ be independent random variables with finite variances, and let $U = X+Y$ and $V = X Y$. Under what condition are $U$ and $V$ uncorrelated?

Let $X$ and $Y$ be independent continuous random variables, and let $U$ be independent of $X$ and $Y$ taking the values $\pm 1$ with probability $\frac{1}{2}$. Define $S = U X$ and $T = U Y$. Show that $S$ and $T$ are in general dependent, but $S^{2}$ and $T^{2}$ are independent.

Let $X, Y, Z$ be independent and identically distributed continuous random variables. Show that $\mathbb {P}\left(X > Y\right) = \mathbb {P}\left(Z > Y\right) = \frac{1}{2}$. What is $\mathbb {P}\left(Z > Y \mid X > Y\right)$?

Let $(X, Y)$ and $(U, V)$ be independent random vectors with common distribution function $F(x, y)$ and marginal distribution functions $F_{X}(x)$ and $F_{Y}(y)$. Show that, when $\left|\operatorname {Cov}\left[X, Y\right]\right| < \infty$, we have that $\mathbb {E}\left[(X-U)(Y-V)\right] = 2 \operatorname {Cov}\left[X, Y\right]$ and

Let the pair $X, Y$ have the bivariate normal distribution with means $0$, variances $1$, and correlation $\rho$. Show that

Let $X$ be exponentially distributed with parameter $\lambda$. Let $N$ be the greatest integer not greater than $X$, and set $M = X-N$. Show that $M$ and $N$ are independent. Find the density function of $M$ and the distribution of $N$.

Let $X$ and $Y$ be independent $N(0,1)$ random variables. Show, if $2 a < 1,2 b < 1$, and $4 b^{2} < (1-2 a)(1-2 c)$, that

Let $X, Y$ be random variables with joint density function $f(x, y)$ and marginal densities $g, h$. Their contingency coefficient is given as

If $X, Y$ are bivariate normal with correlation $\rho \in (-1,1)$, show that $1+\phi^{2} = 1 /\left(1-\rho^{2}\right)$. Show in addition that their mutual information (see Exercise (3.6.5)) is $I = -\frac{1}{2} \log \left(1-\rho^{2}\right)$.

Let $X$ be uniformly distributed on $[-1,1]$. Are the random variables $Z_{n} = \cos (n \pi X), n = 1,2, \ldots$, correlated? Are they independent? Explain your answers.

The mean absolute difference, or MAD, of two independent random variables $X, Y$, with common distribution function $F$, is given by $\operatorname {MAD} = \mathbb {E}\left[\left|X-Y\right|\right]$. Show that, for non-negative random variables $X, Y$,

More generally, for $X$ and $Y$ taking values in $\mathbb {R}$, with a common continuous distribution function with unique inverse $Q$, show that

Find $\operatorname {MAD}$ for the following distributions:

Let $X, Y$ be independent $N(0,1)$ random variables, and $U = \min \left\{ \left|X\right|,\left|Y\right|\right\}$, $V = \max \left\{ \left|X\right|,\left|Y\right|\right\}$. Show that $\mathbb {E}\left[U / V\right] = (4 / \pi ) \log \sqrt{2}$.

A point $P = (X, Y)$ is selected uniformly at random inside a triangle $\Delta$ with corners $(1,0),(1,1),(0,1)$. Show that a triangle ABC with $\mathrm{BC} = X, \mathrm{CA} = Y, \mathrm{AB} = 2-X-Y$ can always be constructed, and prove that the angle $\widehat{\mathrm{ABC}}$ is obtuse with probability $3-4 \log 2$.

Random variables $X, Y$ have joint density function $f(x, y) = 3\left((x+y)-\left(x^{2}+y^{2}\right)\right)$ for $x, y \in [0,1]$. Show that $X$ and $Y$ are uncorrelated but dependent.