Math Problems Vault

Problem 16.1

Let the random variable $\xi$ take only nonnegative integer values, and let $\psi (z)$ , $\left|z\right| \leqslant 1$ , be its generating function (that is, $\psi (z) = \sum_{n=0}^\infty z^n \mathbb {P}\left\{ \xi =n\right\} = \mathbb {E}\left[z^\xi \right]$ ). Does the generating function $\psi (z)$ uniquely determine the distribution of the random variable $\xi$ ?

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Problem 16.2

For any two independent random variables $\xi$ and $\eta$ , prove that $\psi_{\xi +\eta }(z) = \psi_\xi (z)\psi_\eta (z)$ .

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Problem 16.3

(a)(b)(c)

Let $\xi$ be a nonnegative integer-valued random variable with generating function $\psi (z)$ . Find the generating functions of the quantities:

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(a)

$\xi + 1$ ;

(b)

$2\xi$ ;

(c)

$3\xi + 5$ .

Problem 16.4

(a)(b)(c)(d)

Find the generating functions of random variables having:

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(a)

a Bernoulli distribution with parameter $p$ ;

(b)

a binomial distribution with parameters $p$ and $n$ ;

(c)

a Poisson distribution with parameter $\lambda$ ;

(d)

a geometric distribution with parameter $p$ .

Problem 16.5

(a)(b)(c)(d)

Let $\xi$ be a nonnegative integer-valued random variable with generating function $\psi_\xi$ . In terms of $\psi_\xi$ , express the generating functions of the following sequences:

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(a)

$\mathbb {P}\left\{ \xi \leqslant n\right\}$ ;

(b)

$\mathbb {P}\left\{ \xi \geqslant n\right\}$ ;

(c)

$\mathbb {P}\left\{ \xi < n\right\}$ ;

(d)

$\mathbb {P}\left\{ \xi > n\right\}$ .

Problem 16.6

Let $\left\{ \xi_n\right\}$ be a sequence of nonnegative integer-valued independent identically distributed random variables with generating function $\psi_\xi$ . Let $\nu$ be a positive integer-valued random variable, independent of $\left\{ \xi_n\right\}$ , with generating function $\psi_\nu$ . Find the generating function of the sum of a random number of terms $\xi_1 + \ldots + \xi_\nu$ .

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Problem 16.7

(a)(b)(c)(d)

Find the distribution laws corresponding to the following generating functions:

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(a)

$\dfrac {(1+z)^2}{4}$ ;

(b)

$e^{5(z-1)}$ ;

(c)

$\dfrac {1}{2(1 - z/2)}$ ;

(d)

$(1/3 + 2z/3)^n$ , $n \in \mathbb {N}$ .

Problem 16.8

Let the random variable $\xi$ have generating function $\psi$ . Find the distribution of $\xi$ , if $\psi (1/2^n) = 1/2^n$ for every $n$ .

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Problem 16.9

Prove that the function $\psi (z) = \left|z\right|$ cannot be the generating function of a probability distribution.

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Problem 16.10

Let the random variable $\xi$ take the values $0, 1, \ldots , n-1$ with probability $1/n$ each. Prove that if $n$ is composite, then $\xi$ can be represented as the sum of independent integer-valued random variables.

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Problem 16.11

Let the random variable $\xi$ take the values 0 and 1 with probability $1/2$ each, and let $\eta$ take the values 0, 1, 2, and 3 with probabilities $1/8$ , $1/4$ , $1/2$ , and $1/8$ respectively. Prove that there does not exist a random variable $\zeta$ , independent of $\xi$ , such that $\xi + \zeta = \eta$ .

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Problem 16.12

Solve problem 10.39 using generating functions.

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Problem 16.13

Let $\left\{ \xi_n\right\}$ be a sequence of independent identically distributed random variables with common distribution function $F$ . Let $\nu$ be a positive integer-valued random variable, independent of $\left\{ \xi_n\right\}$ , with generating function $\psi$ . Prove that the distribution function of the random variable $\max (\xi_1, \ldots , \xi_\nu )$ equals $\psi (F(x))$ .

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