Conditional distributions and conditional expectation

A point is picked uniformly at random on the surface of a unit sphere. Writing $\Theta$ and $\Phi$ for its longitude and latitude, find the conditional density functions of $\Theta$ given $\Phi$, and of $\Phi$ given $\Theta$.

Show that the conditional expectation $\psi (X) = \mathbb {E}\left[Y \mid X\right]$ satisfies $\mathbb {E}\left[\psi (X) g(X)\right] = \mathbb {E}\left[Y g(X)\right]$, for any function $g$ for which both expectations exist.

Construct an example of two random variables $X$ and $Y$ for which $\mathbb {E}\left[Y\right] = \infty$ but such that $\mathbb {E}\left[Y \mid X\right] < \infty$ almost surely.

Find the conditional density function and expectation of $Y$ given $X$ when they have joint density function:

Let $Y$ be distributed as $\operatorname {bin}(n, X)$, where $X$ is a random variable having a beta distribution on $[0,1]$ with parameters $a$ and $b$. Describe the distribution of $Y$, and find its mean and variance. What is the distribution of $Y$ in the special case when $X$ is uniform?

Let $\left\{ X_{r}: r \geq 1\right\}$ be independent and uniformly distributed on $[0,1]$. Let $0 < x < 1$ and define

Show that $\mathbb {P}\left(N > n\right) = x^{n} / n!$, and hence find the mean and variance of $N$.

Let $X$ and $Y$ be random variables with correlation $\rho$. Show that $\mathbb {E}\left[\operatorname {Var}\left[Y \mid X\right]\right] \leq \left(1-\rho^{2}\right) \operatorname {Var}\left[Y\right]$.

Let $X, Y, Z$ be independent and exponential random variables with respective parameters $\lambda , \mu , \nu$. Find $\mathbb {P}\left(X < Y < Z\right)$.

Let $X$ and $Y$ have the joint density $f(x, y) = c x(y-x) e^{-y}, 0 \leq x \leq y < \infty$.

Let $\left\{ X_{r}: r \geq 0\right\}$ be independent and identically distributed random variables with density function $f$ and distribution function $F$. Let $N = \min \left\{ n \geq 1: X_{n} > X_{0}\right\}$ and $M = \min \left\{ n \geq 1 : X_{0} \geq X_{1} \geq \cdots \geq X_{n-1} < X_{n}\right\}$. Show that $X_{N}$ has distribution function $F+(1-F) \log (1-F)$, and find $\mathbb {P}\left(M = m\right)$.

Let the point $P = (X, Y)$ be uniformly distributed in the square $S = [0,1]^{2}$, and denote by $f(x \mid D)$ the density function of $X$ conditional on the event $D$ that $P$ lies on the diagonal of $S$ through the origin.

Let $X_{1}, X_{2}, \ldots$ be independent, each with density function $f: \mathbb {R} \rightarrow [0, \infty )$. The index $r > 1$ is called a record time if $X_{r} > \max \left\{ X_{1}, X_{2}, \ldots , X_{r-1}\right\}$, and $r = 1$ is called a record time by convention. Let $A_{r}$ be the event that $r$ is a record time.

A unit stick is broken uniformly at random, and the larger piece is broken again uniformly at random. Show that the probability the three pieces may be used to form a triangle is $2 \log 2-1$

Let $U_{1}, U_{2}, \ldots , U_{n+1}$ be independent random variables with the uniform distribution on $[0,1]$, and let

Show that $\mathbb {E}\left[U_{(1)}\right] = 1 /(n+1)$ and deduce that