Functions of random variables

Let $X, Y$, and $Z$ be independent and uniformly distributed on $[0,1]$. Find the joint density function of $X Y$ and $Z^{2}$, and show that $\mathbb {P}\left(X Y < Z^{2}\right) = \frac{5}{9}$.

Let $X$ and $Y$ be independent exponential random variables with parameter 1. Find the joint density function of $U = X+Y$ and $V = X /(X+Y)$, and deduce that $V$ is uniformly distributed on $[0,1]$.

Let $X$ be uniformly distributed on $\left[0, \frac{1}{2} \pi \right]$. Find the density function of $Y = \sin X$.

Find the density function of $Y = \sin^{-1} X$ when:

Let $X$ and $Y$ have the bivariate normal density function

Show that $X$ and $Z = (Y-\rho X) / \sqrt{1-\rho^{2}}$ are independent $N(0,1)$ variables, and deduce that

Let $X$ and $Y$ have the standard bivariate normal density function of Exercise (4.7.5), and define $Z = \max \left\{ X, Y\right\}$. Show that $\mathbb {E}\left[Z\right] = \sqrt{(1-\rho ) / \pi }$, and $\mathbb {E}\left[Z^{2}\right] = 1$.

Let $X$ and $Y$ be independent exponential random variables with parameters $\lambda$ and $\mu$. Show that $Z = \min \left\{ X, Y\right\}$ is independent of the event $\left\{ X < Y\right\}$. Find:

A point $(X, Y)$ is picked at random uniformly in the unit circle. Find the joint density of $R$ and $X$, where $R^{2} = X^{2}+Y^{2}$.

A point $(X, Y, Z)$ is picked uniformly at random inside the unit ball of $\mathbb {R}^{3}$. Find the joint density of $Z$ and $R$, where $R^{2} = X^{2}+Y^{2}+Z^{2}$.

Let $X$ and $Y$ be independent and exponentially distributed with parameters $\lambda$ and $\mu$. Find the joint distribution of $S = X+Y$ and $R = X /(X+Y)$. What is the density of $R$?

Find the density of $Y = a /\left(1+X^{2}\right)$, where $X$ has the Cauchy distribution.

Let $(X, Y)$ have the bivariate normal density of Exercise (4.7.5) with $0 \leq \rho < 1$. Show that

where $c = (b-\rho a) / \sqrt{1-\rho^{2}}, d = (a-\rho b) / \sqrt{1-\rho^{2}}$, and $\phi$ and $\Phi$ are the density and distribution function of the $N(0,1)$ distribution.

Let $X$ have the Cauchy distribution. Show that $Y = X^{-1}$ has the Cauchy distribution also. Find another non-trivial distribution with this property of invariance.

Let $X$ and $Y$ be independent and gamma distributed as $\Gamma (\lambda , \alpha ), \Gamma (\lambda , \beta )$ respectively.

Let $X, Y$ be independent, positive, continuous random variables, such that $1-F_{Y}(y) = \left(1-F_{X}(y)\right)^{\lambda }$ for $y > 0$. The positive parameter $\lambda$ may be called the frailty or proportional hazard. In more general models, $\lambda$ may itself be a random variable.

Show that

Let $X$ and $Y$ be independent random variables, where $X$ has an arc sine distribution and $Y$ a Rayleigh distribution:

Write down the joint density function of the pair $(Y, X Y)$, and deduce that $X Y$ has the standard normal distribution.

Let $U$ be uniformly distributed on the interval $(0,1)$.

Let $(X, Y)$ have the standard bivariate normal distribution with correlation $\rho$.

Let $(X, Y)$ have the standard bivariate normal distribution with correlation $\rho$. Show that

where $\phi$ and $\Phi$ are the $N(0,1)$ density and distribution functions.

Let $(X, Y)$ have the standard bivariate normal distribution with density function $f$ and correlation $\rho$. Show that the probability that $(X, Y)$ lies in the interior of the ellipse $f(x, y) = k$ is $1-\exp \left\{ -A /\left(2 \pi \sqrt{1-\rho^{2}}\right)\right\}$, where $A$ is the area of the ellipse.

Let $R = X / Y$ where $X, Y$ are independent and uniformly distributed on $(0,1)$. Find the probability that the integer closest to $R$ is odd.

Let $(X, Y)$ have the standard bivariate normal distribution with correlation $\rho$ and density function $f$. Show that $f(X, Y)$ is uniformly distributed on the interval $(0, \zeta )$ where $\zeta = 1 /\left\{ 2 \pi \sqrt{1-\rho^{2}}\right\}$.

A number $n$ of friends each visit Old Slaughter's coffee house independently and uniformly at random during their lunch break from noon to 1 pm. Each leaves after $\delta$ hours (or at 1 pm if that is sooner), where $\delta < 1 /(n-1)$. Show that the probability that none of them meet inside is $(1-(n-1) \delta )^{n}$.

Let $X$ and $Y$ be independent random variables with the uniform distribution on $(0,1)$. Find the joint density function of $W = X Y$ and $Z = Y / X$, and deduce their marginal density functions.

Let $X$ have the $N(0,1)$ distribution, and let $G$ be a function with derivative $g$. Show that

The joint density function of the pair $(X, Y)$ is

Find the joint density function of the pair $U = X$, and $V = X+Y$, and deduce the density function of $V$.