Math Problems Vault

Problem 2.1

Two persons shoot at a target, each has $n$ shots. Using generating functions, find the probability $p$ that they have the same number of successes, given that each of them has a probability $0.5$ of success (hitting the target) at each shot.

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Problem 2.2

(a)(b)

To obtain the title of champion in chess, an opponent has to win at least $12.5$ points out of the maximum of $24$ . In the case of a tie ( $12:12$ ), the defending champion maintains the title. The probability that each of the two players wins a game is the same, and equals half the probability of a tie. Find:

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(a)

the probability $p_{\mathrm{ch}}$ that the defending champion will maintain his title, as well as the probability $p_{\mathrm{op}}$ that the opponent will become the champion;

(b)

the probability $p$ that $20$ games will be played in the contest.

Problem 2.3

In a lottery, tickets are numbered from $000000$ to $999999$ and they are all equally probable. What is the probability $p$ that the sum of the digits of a chosen number be $21$ ?

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Problem 2.4

Prove that the Pascal distribution, for $r \to \infty$ and $rq \to \lambda$ , converges to the Poisson with parameter $\lambda$ .

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Problem 2.5

Let $X, Y, Z$ be independent random variables, each taking the values $1, 2, \ldots , n$ with the same probability (uniformly). Find $\mathbb {P}\left(X + Y = 2Z\right)$ from their joint generating function $G(t_{1}, t_{2}, t_{3})$ .

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Problem 2.6

In Exercise 315, let

q_{n,k} = \mathbb {P}\left(S_{n} \leq k\right).

Find the generating function $Q(t)$ of $q_{n,k}$ for a given $n$ and hence $q_{n,k}$ .

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Problem 2.7

(Continuation of Problem 2.6.) Suppose $k$ and $m$ tend to infinity so that $k/m \to x$ . Find the limit of $q_{n,k}$ .

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Problem 2.8

Find the densities of the distributions with characteristic functions

\varphi _{1}(t) = \frac{1+it}{1+t^{2}}, \qquad \varphi _{2}(t) = \frac{1-it}{1+t^{2}}.

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Problem 2.9

Find the moments of the Laplace distribution with characteristic function

\varphi (t) = \frac{1}{1+t^{2}}.

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Problem 2.10

Determine the distribution with characteristic function

\varphi (t) = \frac{1}{2e^{-it} - 1}.

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Problem 2.11

Find the characteristic function of $Y = aF(X) + b$ , where $X$ is a continuous random variable and $F$ is its distribution function.

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Problem 2.12

A vector $\xi = (X_{1}, \ldots , X_{n})$ follows the $n$ -dimensional normal distribution with $\mathbb {E}\left[X_{i}\right] = a$ , $\operatorname {Var}\left[X_{i}\right] = \sigma^{2}$ $(i = 1, \ldots , n)$ and covariances

\operatorname {Cov}\left[ X_{i}, X_{i+1} \right] = \rho \sigma ^{2}, \qquad i = 1, 2, \ldots , n-1,

\operatorname {Cov}\left[ X_{i}, X_{j} \right] = 0 \qquad \text{for} \quad \left|i-j\right| > 1.

Find the characteristic function of $\xi$ .

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Problem 2.13

(a)(b)

Let $\varphi (t)$ be the characteristic function of a random variable $X$ with $\mathbb {E}\left[X^{2}\right] = \mu_{2}'$ .

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(a)

Prove that

\psi (t) = -\frac{1}{\mu _{2}'}\varphi ''(t)

is a characteristic function.

(b)

Conclude that

\psi _{0}(t) = (1-t^{2})e^{-t^{2}/2}

is a characteristic function.

Problem 2.14

Show that

a(t) = \begin{cases} 1 - (\left|t\right|/\pi ), & \text{for } \left|t\right| < \pi , \\ 0, & \text{for } \left|t\right| > \pi , \end{cases}

is the characteristic function of a random variable $X$ , and determine the corresponding distribution.

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Problem 2.15

Let $X_{1}, X_{2}, X_{3}$ be independent $\mathscr {N}\left(0, 1\right)$ . Find the characteristic function of the pair $Z_{1}, Z_{2}$ , where

Z_{1} = X_{2}X_{3}, \qquad Z_{2} = X_{1}X_{3}.

Hence conclude the characteristic function of $Z_{1}$ . Are $Z_{1}, Z_{2}$ independent?

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Problem 2.16

Find the distributions corresponding to the characteristic functions

\varphi _{1}(t) = \frac{1}{\cosh t}, \qquad \varphi _{2}(t) = \frac{1}{\sinh t}, \qquad \varphi _{3}(t) = \frac{1}{\cosh ^{2} t}.

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Problem 2.17

Show that

P(s_{1}, s_{2}) = p_{0}^{r}(1 - p_{1}s_{1} - p_{2}s_{2})^{-r},

with positive parameters and $p_{0} + p_{1} + p_{2} = 1$ is the probability generating function of a pair $(X, Y)$ , so that the marginal distributions of $X$ and $Y$ and $X + Y$ are negative binomials.

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Problem 2.18

Let a ``round'' consist of a sequence of Bernoulli trials before the first failure. Find the generating function and the probability function of the total number $S_{k}$ of successes in $k$ rounds.

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Problem 2.19

(Continuation of Problem 2.18.) Let $G$ be the number of successive rounds up to the $n$ th success (i.e., the $n$ th success occurs during the $G$ th round). Find the distribution of $G$ and the

\mathbb {E}\left[G\right], \operatorname {Var}\left[G\right].

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Problem 2.20

(Continuation of Problem 2.19.) Consider two independent sequences of Bernoulli variables with parameters $p_{1}$ and $p_{2}$ , respectively, or two players with ``skills'' $p_{1}$ and $p_{2}$ , respectively. Prove that each player will need the same number of rounds (trials) until the $N$ th success (a tie) with probability

(p_{1}p_{2})^{N} \sum _{v=1}^{\infty } \binom {N+v-2}{v-1}(q_{1}q_{2})^{v-1}

= (p_{1}p_{2})^{N}(1 - q_{1}q_{2})^{1-2N} \sum _{k=0}^{N-1} \binom {N-1}{k}(q_{1}q_{2})^{k}.

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Problem 2.21

Show that the characteristic function of the $\beta$ density (2.10) is

\frac{\Gamma (p+q)}{\Gamma (p)} \sum _{j=0}^{\infty } \frac{\Gamma (p+j)(it)^{j}}{\Gamma (p+q+j)\Gamma (j+1)}.

Formula (2.10), the beta density on $(0,1)$ with parameters $p, q > 0$ :

f(x) = \frac{\Gamma (p+q)}{\Gamma (p)\Gamma (q)} x^{p-1}(1-x)^{q-1}, \qquad 0 < x < 1.

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Problem 2.22

Show that the characteristic function of the density in Problem 1.13 is of the form $\varphi (t_{1}^{2} + t_{2}^{2} + \cdots + t_{n}^{2})$ (see Problem 1.21). If

\varphi (t_{1}^{2} + \cdots + t_{n}^{2}) = \prod _{i=1}^{n} \varphi (t_{i}^{2}), \tag {*}

that is, $X_{1}, X_{2}, \ldots , X_{n}$ are independent, prove that $\overrightarrow {X}$ is $\mathscr {N}\left(0, I\right)$ . In other words, a spherically symmetric random vector has independent components if and only if it is multivariate normal.

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