Math Problems Vault

Problem 171

(a)(b)

The joint distribution of $(X, Y)$ is defined by $\mathbb {P}\left(X=0, Y=0\right) = \mathbb {P}\left(X=0, Y=1\right) = \mathbb {P}\left(X=1, Y=1\right) = 1/3$ .

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(a)

Find the marginal distribution functions of $X$ , $Y$ .

(b)

Examine whether the points $P_{1}: (-1/2, 0)$ , $P_{2}: (0, 1)$ are continuity points of $F(x, y)$ .

Problem 172

Show that the function

F(x, y) = \begin{cases} 0 & \text{for } x + y < 1, \\ 1 & \text{for } x + y \geq 1, \end{cases}

is not a joint distribution function.

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Problem 173

We consider a family with two children. Let $X_{n} = 1$ if the $n$ th child is a boy for $n = 1, 2$ and $X_{3} = 1$ if there is only one boy, otherwise $X_{i} = 0$ ( $i = 1, 2, 3$ ). Show that the $X_{i}$ are pairwise independent but not completely independent.

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Problem 174

(a)(b)

$X$ and $Y$ have the joint density

f(x, y) = c x^{n_{1}-1} (y-x)^{n_{2}-1} e^{-y} \quad \text{for } 0 < x < y < \infty .

Find

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(a)

the constant $c$ ,

(b)

the marginal distributions of $X$ and $Y$ .

Problem 175

The $k$ -dimensional hypergeometric distribution has probability function

\mathbb {P}\left(X_{1}=n_{1}, \ldots , X_{k}=n_{k}\right) = \binom {N p_{1}}{n_{1}} \cdots \binom {N p_{k+1}}{n_{k+1}} \bigg/ \binom {N}{n},

where

n_{k+1} = n - (n_{1}+n_{2}+\cdots +n_{k}), \qquad p_{k+1} = 1-(p_{1}+p_{2}+\cdots +p_{k}), \quad n_{i} > 0, \; p_{i} > 0.

Show that the conditional distribution of $X_{k}$ given $X_{1}, \ldots , X_{k-1}$ is hypergeometric.

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Problem 176

If $X_{1}, X_{2}, \ldots , X_{k}$ are distributed according to the multinomial distribution, the conditional distribution of $X_{1}$ , given $X_{2}=n_{2}, \ldots , X_{k}=n_{k}$ , is binomial with parameters $n-(n_{2}+\cdots +n_{k})$ and $p_{1}/(p_{1}+p_{k+1})$ .

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Problem 177

If $X_{1}, X_{2}, \ldots , X_{k}$ have a Dirichlet distribution, show that the conditional distribution of $X_{k}/S_{k}$ , for $X_{i}=x_{i}$ , where $S_{k} = 1-(X_{1}+\cdots +X_{k-1})$ is $\beta (n_{k}, n_{k+1})$ .

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Problem 178

(a)(b)

Let $X = (X_{1}, X_{2}, X_{3})$ be uniformly distributed in the subset of the positive orthant defined by $x_{1} \geq 0$ , $x_{2} \geq 0$ , $x_{3} \geq 0$ , $x_{1}+x_{2}+x_{3} \leq c$ . Find,

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(a)

the density of $X$ ,

(b)

the marginal distribution of $(X_{1}, X_{2})$ .

Problem 179

(a)(b)

A bivariate normal distribution has density

f(x, y) = c \exp \left[-x^{2}+xy-y^{2}\right].

Recall that the $n$ -dimensional (nonsingular) normal distribution $N(\mu , \Sigma )$ has density

f(x) = \left[(2\pi )^{p} \left|\Sigma \right|\right]^{-1/2} \exp \left[-\tfrac {1}{2}(x-\mu )' \Sigma ^{-1} (x-\mu )\right],

where $\mu = (\mu_{1}, \ldots , \mu_{n})' = \mathbb {E}\left[X\right]$ denotes the mean vector and $\Sigma$ denotes the (positive definite) covariance matrix of the normal random vector $X$ .

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(a)

Find the constant $c$ and the moments of order 2.

(b)

Find the curves of constant density $f(x, y) = c^{*}$ .

Problem 180

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

If $(X, Y)$ is uniform in the triangle $x \geq 0$ , $y \geq 0$ , $x+y \leq 2$ , find,

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(a)

the density of $(X, Y)$ ,

(b)

the density of $X$ ,

(c)

the conditional density of $Y$ for $X = x$ ,

(d)

$\mathbb {E}\left[ Y \mid X=x \right]$ .

Problem 181

(a)(b)(c)

A die is thrown 12 times.

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(a)

Find the probability that every face appears twice.

(b)

Let $X$ be the number of appearances of 6 and $Y$ the number of appearances of 1. Find the joint distribution of $X$ , $Y$ .

(c)

Find $\operatorname {Cov}\left[ X, Y \right]$ .

Problem 182

Show that the most probable value $(n_{1}^{0}, n_{2}^{0}, \ldots , n_{k+1}^{0})$ of the multinomial distribution satisfies the relations

n p_{i} - 1 < n_{i}^{0} \leq (n+k) p_{i}, \qquad i = 1, 2, \ldots , k+1.

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Remark.

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Show that

p_{i} n_{j}^{0} \leq p_{j}(n_{i}^{0}+1) \quad \text{for } i \neq j, \; i, j = 1, 2, \ldots , k+1.

Problem 183

If the number of trials $n$ is large and the $p_{i}$ small so that the $np_{i} = \lambda_{i}$ are moderate, show that the multinomial distribution can be approximated by the so-called multiple Poisson distribution

e^{-(\lambda _{1}+\lambda _{2}+\cdots +\lambda _{k})} \frac{\lambda _{1}^{n_{1}} \lambda _{2}^{n_{2}} \cdots \lambda _{k}^{n_{k}}}{n_{1}! \, n_{2}! \cdots n_{k}!}.

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Problem 184

(a)(b)(c)

Given that the density of $X$ and $Y$ is

f(x, y) = \frac{2}{(1+x+y)^{3}}, \qquad x > 0, \; y > 0,

find,

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(a)

$F(x, y)$ ,

(b)

$f_{X}(x)$ ,

(c)

$f_{Y}(y \mid X=x)$ .

Problem 185

Find the density function $f(x, y)$ of the uniform distribution in the circle $x^{2}+y^{2} \leq 1$ . Find the marginal distributions of $X$ and $Y$ . Are the variables $X$ and $Y$ independent?

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Problem 186

The joint density of the variables $X, Y, Z$ is

f(x, y, z) = 8xyz, \qquad 0 < x, y, z < 1.

Find $\mathbb {P}\left(X < Y < Z\right)$ .

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Problem 187

(a)(b)

For each of the following densities $f(x, y)$ , find $F(x, y)$ , $F_{X}(x)$ , $F_{Y}(y)$ , $f_{X}(x)$ , $f_{Y}(y)$ , $f_{X}(x \mid Y=y)$ , $f_{Y}(y \mid X=x)$ .

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(a)

$f(x, y) = 4xy, \quad 0 < x, \; y < 1$ ,

(b)

$f(x, y) = \tfrac {1}{8}(x^{2}-y^{2}) e^{-x}, \quad 0 < x < \infty , \; \left|y\right| < x$ .

Problem 188

(a)(b)

The joint probability function $p(x_{i}, y_{j})$ of $X, Y$ is given in the following table:

\begin{array}{c|cccc|c}& y = -1 & 0 & 1 & 2 & p(x_{j}) \\ \hline x = -1 & 0.1 & 0.15 & 0 & 0.1 & 0.35 \\ 0 & 0.15 & 0 & 0.1 & 0.2 & 0.45 \\ 1 & 0.05 & 0.05 & 0 & 0.1 & 0.20 \\ \hline q(y_{j}) & 0.3 & 0.2 & 0.1 & 0.4 & 1.00 \end{array}

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(a)

Calculate the regression lines of $X$ on $Y$ and of $Y$ on $X$ .

(b)

If it is possible to observe only $X$ , what is the best estimate of $Y$ in terms of $X$ in the sense of the mean square error?

Problem 189

Show that the discrete variables $X$ and $Y$ with joint probability function $p_{ij} = \mathbb {P}\left(X=x_{i}, Y=y_{j}\right)$ are independent if and only if the matrix of probabilities $P = (p_{ij})$ has rank 1.

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Problem 190

Show that two orthogonal (uncorrelated) variables each taking two values are independent.

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