Generating functions

Find the generating functions of the following mass functions, and state where they converge. Hence calculate their means and variances.

Let $X( \geq 0)$ have probability generating function $G$ and write $t(n) = \mathbb {P}\left(X > n\right)$ for the 'tail' probabilities of $X$. Show that the generating function of the sequence $\left\{ t(n): n \geq 0\right\}$ is $T(s) = (1-G(s)) /(1-s)$. Show that $\mathbb {E}\left[X\right] = T(1)$ and $\operatorname {Var}\left[X\right] = 2 T^{\prime }(1)+T(1)-T(1)^{2}$.

Let $G_{X, Y}(s, t)$ be the joint probability generating function of $X$ and $Y$. Show that $G_{X}(s) = G_{X, Y}(s, 1)$ and $G_{Y}(t) = G_{X, Y}(1, t)$. Show that

Find the joint generating functions of the following joint mass functions, and state for what values of the variables the series converge.

A coin is tossed $n$ times, and heads turns up with probability $p$ on each toss. Assuming the usual independence, show that the joint probability generating function of the numbers $H$ and $T$ of heads and tails is $G_{H, T}(x, y) = \left\{ p x+(1-p) y\right\}^{n}$. Generalize this conclusion to find the joint probability generating function of the multinomial distribution of Exercise (3.5.1).

Let $X$ have the binomial distribution $\operatorname {bin}(n, U)$, where $U$ is uniform on $(0,1)$. Show that $X$ is uniformly distributed on $\left\{ 0,1,2, \ldots , n\right\}$.

Show that

is the joint generating function of four variables that are pairwise and triplewise independent, but are nevertheless not independent.

Let $p_{r} > 0$ and $a_{r} \in \mathbb {R}$ for $1 \leq r \leq n$. Which of the following is a moment generating function, and for what random variable?

Let $G_{1}$ and $G_{2}$ be probability generating functions, and suppose that $0 \leq \alpha \leq 1$. Show that $G_{1} G_{2}$, and $\alpha G_{1}+(1-\alpha ) G_{2}$ are probability generating functions. Is $G(\alpha s) / G(\alpha )$ necessarily a probability generating function?

Let $X_{1}, X_{2}, \ldots$ be independent, continuous random variables with common distribution function $F$, that are independent of the positive integer-valued random variable $Z$. Define the maximum $M = \max \left\{ X_{1}, X_{2}, \ldots , X_{Z}\right\}$. Show that

where $G$ is the probability generating function of $Z$.

Let $0 < p = 1-q < 1$, and let $X$ have the geometric mass function $f(x) = q^{x-1} p$ for $x = 1,2, \ldots$. Find the probability generating function of $Y = \min \left\{ n, X\right\}$ for fixed $n \geq 1$, and show that $\mathbb {E}\left[Y\right] = \left(1-q^{n}\right) / p$.

Let $X_{i}$ be the number of balls in the $i$th of a sequence of bins, and assume the $X_{i}$ are independent with common probability generating function $G_{X}$. There are $N$ such bins, where $N$ is independent of the $X_{i}$ with probability generating function $G_{N}$. Each ball is 'unmarked' with probability $u$, and marked otherwise, with marks appearing independently.