Generating Functions. Characteristic Functions

Find the probability generating function and hence the mean and variance of the following distributions: (a) binomial, (b) Poisson, (c) geometric, (d) Pascal. Moreover, deduce the corresponding moment generating functions and characteristic functions.

Let $X_{1} = X_{2} = X$ when $X$ has the Cauchy density

Prove that

while $X_{1}$, $X_{2}$ are not independent.

From the probability generating function $P_{X}(t)$ of $X = 0, 1, 2, \ldots$, find the probability generating function of $Y = 3X+2$.

Using the probability generating function prove that the random variable $S_{N}$ of formula (6.3) --- $S_{N} = X_{1}+X_{2}+\cdots +X_{N}$, where the random variable $N = 0, 1, \ldots$ and the $X_{i}$ are completely independent and identically distributed --- has variance given by $\operatorname {Var}\left[S_{N}\right] = \mathbb {E}\left[N\right]\operatorname {Var}\left[X\right] + \operatorname {Var}\left[N\right]\mathbb {E}\left[X\right]^{2}$.

A woman continues to have children until she has a boy! Suppose that the probability of having a blond child is $p$; find the probability that the woman will have $k$ blond children. (Assume that the color is independent of sex.)

The number $N$ of visitors to a shop has the Poisson distribution. On average, 150 women visit the shop every day. Of these, $100p_{1}\%$ buy mini, $100p_{2}\%$ buy midi, $100p_{3}\%$ buy maxi, and the remaining buy nothing. (No woman buys more than one dress!!) Let $X_{1}, X_{2}, X_{3}$ be the corresponding numbers of mini, midi, and maxi that are bought in a day. Find the joint probability function of $X_{1}, X_{2}, X_{3}$. Are they independent?

Let $X$, $Y$ be independent with densities

Prove that $\varphi_{X}(t) = \varphi_{Y}(t)$ for $|t| \leq \min \left\{ 1/a, 1/b, 1/c\right\}$, $2c = a+b$.

Let $X_{1}, X_{2}, \ldots , X_{n}$ be independent random variables having the same normal distribution with mean $\mu$ and variance $1$. We set

Moreover, let the random variable $T = aY_{v}$ where $a, v$ are suitable constants, and $Y_{v}$ follows the $\chi^{2}$ distribution with $v$ degrees of freedom. Using characteristic functions, find the constants $a$ and $v$ so that the random variables $S_{n}$ and $T$ have the same means and the same variances.

Prove that $S_{N}$ of formula (6.3) --- $S_{N} = X_{1}+X_{2}+\cdots +X_{N}$, where the random variable $N = 0, 1, \ldots$ and the $X_{i}$ are completely independent and identically distributed --- has the characteristic function

Prove that $\varphi (t) = \exp [\lambda (e^{-|t|}-1)]$ is the characteristic function of a Poisson distribution compounded with (generalized by) a Cauchy distribution. Then, using the inversion formula, find its density.

Let $Z_{n}$ be a random variable following the binomial distribution with parameters $n$ and $p = \lambda /n$ $(\lambda > 0)$. Moreover, let the random variable $Z$ have a Poisson distribution with parameter $\lambda$. Prove that $\lim_{n\to \infty } \varphi_{Z_{n}}(t) = \varphi_{Z}(t)$ $(t \in R)$, where $\varphi_{Z_{n}}(t)$ and $\varphi_{Z}(t)$ are the characteristic functions of $Z_{n}$ and $Z$, respectively. Then deduce the approximation of the binomial by the Poisson.

Calculate the characteristic function of a Gamma distribution with density

Then deduce the characteristic function of $\chi_{n}^{2}$.

At the $n$th toss of a coin a player receives or pays $2^{-n}$ of a drachma depending on whether heads or tails appears. Let $Y_{n}$ be the gain of the player after $n$ tosses. Show that the distribution of $Y_{n}$ as $n\to \infty$ tends to the uniform distribution in the interval $(-1,1)$. First find the characteristic function of $Y_{n}$.

$X$ has distribution function $F(x)$ which is a mixture of two distribution functions $F_{1}, F_{2}$ as follows:

If $F_{1}$ is $N(\mu_{1}, \sigma_{1}^{2})$ and $F_{2}$ is $N(\mu_{2}, \sigma_{2}^{2})$ find the characteristic function of $F$ and then the $\mathbb {E}\left[X\right]$ and $\operatorname {Var}\left[X\right]$.

Find the second characteristic function (cumulant generating function) $\psi (t)$ of the exponential with density

and prove that the cumulant of order $r$, $\kappa_{r}$, is given by

(Continuation of Problem 205.) Let $Y$ be a discrete random variable defined as follows. For $h > 0$ and $k = 0, 1, 2, \ldots$,

Find the probability generating function of $Y$ and then show that $\mathbb {E}\left[X\right] > \mathbb {E}\left[Y\right]$, $\operatorname {Var}\left[X\right] < \operatorname {Var}\left[Y\right]$.

Suppose that the density $f(x,y) = g(x^{2}+y^{2})$, i.e., it has circular symmetry about the origin. Show that the characteristic function of $f$, $\varphi (t,u)$, is a function of $t^{2}+u^{2}$ (only). Hence show that the only distribution of independent random variables with the above property is the normal. (Consider an orthogonal transformation of $t, u$.)

Consider two independent random samples $X_{1}, X_{2}, \ldots , X_{n}$ and $Y_{1}, Y_{2}, \ldots , Y_{n}$ from the Laplace distribution

Show that the means of the differences $X_{i}-Y_{i}$ and $X_{i}+Y_{i}$ have the same distribution but are not independent. Are they orthogonal?

If $\varphi (t)$ is a characteristic function show that, (a) $|\varphi (t)|^{2}$, (b) $e^{\lambda (\varphi (t)-1)}$ are also characteristic functions.

Calculate the probability generating function of the multinomial distribution and then show that

(Continuation of Problem 210.) If, as $n\to \infty$, $np_{j}\to \lambda_{j}$ $(j=1,2,\ldots ,k+1)$, then the multinomial distribution tends to the $k$-dimensional Poisson $Y=(Y_{1},Y_{2},\ldots ,Y_{k})$ where the $Y_{i}$ are independent Poisson, $Y_{j}$ with parameter $\lambda_{j}$ (cf. Exercise 183).

Calculate the characteristic function of the negative multinomial distribution

where $s>0$, $\theta_{j}>0$ $(n_{j}=0,1,2,\ldots )$, and $n_{k+1}=n-(n_{1}+\cdots +n_{k})$, and hence

$X, Y$ have a distribution of the continuous type with characteristic function $\varphi (t_{1},t_{2})$. Show that the conditional characteristic function of $X$ given $Y=y$ is

A random sample $X_{1}, X_{2}, \ldots , X_{n_{1}+n_{2}-n}$ on a continuous random variable $X$ with $-\infty < X < \infty$, is divided at random into three subsets of $(n_{1}-n)$, $(n_{2}-n)$, and $n$ observations $(n_{1}>n, n_{2}>n)$. Let $S_{1}, S_{2}$, and $S_{3}$ denote the corresponding sums. Find the characteristic function of

Show that the following distributions are infinitely divisible: (a) Pascal, (b) Cauchy, (c) Laplace, (d) Gamma.