Math Problems Vault

Problem 216

(a)(b)

The radius of a circle is approximately measured so that it has the uniform distribution in the interval $(a, b)$ . Find the distribution:

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(a)

of the length of the circumference of the circle;

(b)

of the area of the circle.

Problem 217

Suppose that $X$ and $Y$ are independent random variables with the same exponential density

f(x) = \theta e^{-\theta x}, \qquad x > 0.

Show that the sum $X + Y$ and the ratio $X/Y$ are independent.

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Problem 218

Suppose that $X$ and $Y$ are independent uniform variables in $(0, 1)$ . Find the probability that the roots of $\lambda^{2} + 2X\lambda + Y = 0$ are real.

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Problem 219

The angle $\varphi$ at which a projectile is fired with initial velocity $v$ follows the uniform distribution in the interval $(0, \pi /2)$ . Find the distribution of the horizontal distance $d$ between the firing point and the point at which the projectile falls.

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Problem 220

(a)(b)

Suppose that $X$ and $Y$ are normal variables with mean $0$ , variance $\sigma^{2}$ , and correlation coefficient $\rho$ .

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(a)

Find the distribution of $X/Y$ .

(b)

Using the above result prove that

\mathbb {P}\left(X < 0, Y > 0\right) = \frac{1}{2} \mathbb {P}\left(XY < 0\right) = \frac{1}{4} - \frac{\arcsin \rho }{2\pi }.

Verify this by integrating the density of $(X, Y)$ .

Problem 221

Let $A = (X, Y)$ be the point of impact of a shot on a vertical target with center at origin $O$ and suppose the distribution of $A$ is that of Exercise 220 with $\rho = 0$ (called circular normal). Show that the distance $R = (OA)$ between the point of impact and the centre of the target and the angle of $OA$ with the horizontal axis (i.e., the polar coordinates of $A$ ) are independent variables. Find the distribution of $R^{2}$ .

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Problem 222

(a)(b)

If $X$ has the uniform distribution on the interval $(0, 1)$ , then:

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(a)

Prove that $Y = aX + b$ is also uniformly distributed ( $a$ and $b$ constants).

(b)

Find the distribution of $Y = AX^{2} + BX + C$ ( $A, B, C$ constants).

Problem 223

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

If $X_{1}, X_{2}$ are independent and uniform on the interval $(0, 1)$ , find the densities of:

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(a)

$X_{1} + X_{2}$ ;

(b)

$X_{1} - X_{2}$ ;

(c)

$\left|X_{1} - X_{2}\right|$ ;

(d)

$X_{1}/X_{2}$ .

Problem 224

(a)(b)

Two friends $A$ and $B$ agree to meet between 12 (noon) and 1 p.m. at a restaurant. Supposing that they arrive at random between 12 and 1 p.m. independently of each other and the lunch lasts 30 minutes, what is the probability that they meet in the restaurant? Let $T$ be the instant of their meeting. What is the conditional distribution of $T$ ,

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(a)

given that they meet;

(b)

given that they meet and $A$ arrives first?

Problem 225

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

f_{X}(x) = \frac{1}{\pi } \frac{1}{\sqrt{1 - x^{2}}}, \qquad \left|x\right| < 1, \qquad f_{Y}(y) = \frac{y}{\sigma ^{2}} e^{-y/2\sigma ^{2}}.

Show that $XY$ is $\mathscr {N}\left(0, \sigma^{2}\right)$ .

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Problem 226

(a)(b)

Given $n$ independent random numbers $X_{1}, X_{2}, \ldots , X_{n}$ from $(0, 1)$ , called pseudorandom numbers, show that:

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(a)

$Y = -\sum_{i=1}^{n} 2 \log X_{1}$ has the $\chi^{2}$ distribution with $2n$ degrees of freedom;

(b)

the variables

\xi = -\sqrt{-2 \log X_{1}} \cos (2\pi X_{2}), \qquad \eta = \sqrt{-2 \log X_{1}} \sin (2\pi X_{2}),

are independent $\mathscr {N}\left(0, 1\right)$ .

Thus (a) generates samples of $\chi^{2}$ distributions with an even number of degrees of freedom, while (b) generates a pair of independent normal variables.

Problem 227

Let $X$ be a continuous random variable $X$ with distribution function $F(\cdot )$ ; consider the random variable

Y = F(X) = \int _{-\infty }^{X} f(u) \, du.

This transformation is called the probability integral transformation.

Show that the distribution of $Y$ is uniform in the interval $(0, 1)$ , that is,

F_{Y}(y) = y, \qquad 0 < y < 1.

Given a random sample $X_{1}, X_{2}, \ldots , X_{n}$ from the uniform distribution, show that the solutions $y_{1}, y_{2}, \ldots , y_{n}$ of $y_{1} = F^{-1}(x_{1}), y_{2} = F^{-1}(x_{2}), \ldots , y_{n} = F^{-1}(x_{n})$ form a random sample from the $F$ distribution.

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Problem 228

(Continuation of Problem 227.) Suppose the height $X$ of men has the normal distribution $\mathscr {N}\left(167 \text{ cm}, 9 \text{ cm}^{2}\right)$ and a random sample of 10 men is taken with heights $X_{1}, X_{2}, \ldots , X_{10}$ . What is probability that the (random) interval $\delta = [\min (X_{1}, \ldots , X_{10}), \max (X_{1}, \ldots , X_{10})]$ covers at least 95 of the population of heights? That is, the $\mathbb {P}\left(\mathbb {P}\left(X \in \delta \right) \geq 0.95\right)$ is to be calculated; note that the $\mathbb {P}\left(X \in \delta \right)$ , depending on the random interval $\delta$ , is a random variable.

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Problem 229

(a)(b)(c)

$X$ is called a lognormal variable, if the $\log X = Y$ has a normal distribution $\mathscr {N}\left(\mu , \sigma^{2}\right)$ .

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(a)

Find the density of $X$ .

(b)

Find $E(X)$ and $\operatorname {Var}\left[X\right]$ .

(c)

If the $X_{i}$ are independent lognormal random variables, their product $X_{1} X_{2} \ldots X_{n}$ is also lognormal.

Problem 230

Show that if $F(x, y)$ is the distribution function of $X$ and $Y$ and

Z = \max (X, Y), \qquad W = \min (X, Y),

then

F_{Z}(z) = F(z, z), \qquad F_{W}(w) = F_{X}(w) + F_{Y}(w) - F(w, w).

If $F(x, y)$ is continuous find the densities of $Z$ and $W$ .

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Problem 231

(Continuation of Problem 230.) If $X$ and $Y$ are independent $\mathscr {N}\left(0, 1\right)$ , show that

E\{ \max (X, Y)\} = \frac{1}{\sqrt{\pi }}.

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Problem 232

(a)(b)

Let $X = (X_{1}, X_{2}, \ldots , X_{k})$ have the Dirichlet distribution. Find the distribution,

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(a)

of $Y_{k} = X_{1} + X_{2} + \cdots + X_{k}$ , and

(b)

of $Y_{i} = X_{1} + X_{2} + \cdots + X_{i}$ $(i = 1, 2, \ldots , k)$ .

Problem 233

(Continuation of Problem 232.) If $X_{i}$ $(i = 1, 2, \ldots , k+1)$ are independent Gamma variables with parameters $n_{1}, \ldots , n_{k+1}$ and $\lambda$ , respectively, show that the

Y_{i} = \frac{X_{i}}{X_{1} + \cdots + X_{k+1}}, \qquad i = 1, 2, \ldots , k,

have the $k$ -dimensional Dirichlet distribution (5.11).

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Problem 234

(a)(b)

Let $X$ and $Y$ be independent Poisson variables with parameters $\lambda$ and $\mu$ . Show that:

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(a)

the sum $X + Y$ is also Poisson (use the convolution formula, cf. (7.3));

(b)

the conditional distribution of $X$ given that $X + Y$ is binomial.

Problem 235

(a)(b)

Let $X$ and $Y$ be independent binomial with parameters $N, p$ and $M, p$ , respectively. Show that:

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(a)

the sum of $X + Y$ is binomial (use the convolution formula);

(b)

the conditional distribution of $X$ given $X + Y$ is hypergeometric, independent of $p$ .

Problem 236

(a)(b)

Show that the mean

\overline{X} = \frac{1}{n} \sum _{i=1}^{n} X_{i}

and the variance

s^{2} = \frac{1}{n-1} \sum _{i=1}^{n} (X_{i} - \overline{X})^{2}

of the random sample $X_{1}, X_{2}, \ldots , X_{n}$ from $\mathscr {N}\left(\mu , \sigma^{2}\right)$ are independent. Hence deduce the distribution of $(n-1)s^{2}$ .

Hint:

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(a)

Use any orthogonal transformation $\overrightarrow {Y} = H\overrightarrow {X}$ such that

Y_{n} = \sqrt{n}\overline{X}, \qquad \overrightarrow {Y} = (Y_{1}, \ldots , Y_{n})', \qquad \overrightarrow {X} = (X_{1}, \ldots , X_{n})'.

(b)

$\operatorname {Cov}\left[ \overline{X}, X_{i} - \overline{X} \right] = 0$ .

Show that $\operatorname {Cov}\left[ \overline{X}, s^{2} \right] = \mu_{3}/n$ for samples from nonnormal populations, where $\mu_{3}$ is the population central moment of order 3.

Problem 237

Test for the Behrens–Fisher problem. To test whether two heteroscedastic normal populations have the same means one uses the statistic

t = [m(m-1)]^{1/2} (\overline{x} - \overline{y}) \Big/ \left[\sum _{j=1}^{m} (u_{j} - \overline{u})^{2}\right]^{1/2},

where $\overline{x}, \overline{y}$ are the sample means of the independent samples from $\mathscr {N}\left(\mu_{1}, \sigma_{1}^{2}\right)$ and from $\mathscr {N}\left(\mu_{2}, \sigma_{2}^{2}\right)$ , respectively, $m \leq n$ and $u_{j} = x_{j} - (\sqrt{m/n}) y_{j}$ $(j = 1, 2, \ldots , m)$ . Show that $t$ has Student's distribution with $m-1$ degrees of freedom when $\mu_{1} = \mu_{2} = \mu$ .

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Problem 238

Find the joint density of the ordered statistics $X_{(1)} < X_{(2)} < \cdots < X_{(n)}$ from a distribution of the continuous type.

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Problem 239

(a)(b)

If $X_{1}, X_{2}, \ldots , X_{n}$ is a random sample from the continuous distribution $F$ . Find:

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(a)

the joint density function of the random variables

Y = F\left(\min _{1 \leq i \leq n} X_{i}\right), \qquad Z = F\left(\max _{1 \leq i \leq n} X_{i}\right);

(b)

$E(R) = E(Z - Y)$ where $R$ is the range of the sample.

Problem 240

Let $X_{(1)} < X_{(2)} < \cdots < X_{(n)}$ be the ordered sample from a continuous distribution $F(x)$ . Show that

E[X_{(k+1)} - X_{(k)}] = \binom {n}{k} \int _{-\infty }^{+\infty } F^{n-k}(x) [1 - F(x)]^{k} \, dx.

Hint: Use the joint density of $X_{(k)}, X_{(k+1)}$ .

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Problem 241

(a)(b)

A random sample of size $n$ is taken from a uniform distribution in the interval $(0, 1)$ . Find,

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(a)

the mean $E[X_{(k)}]$ ,

(b)

the distribution of $R = X_{(n)} - X_{(1)}$ .

Problem 242

If $y_{1} < y_{2} < y_{3}$ are the ordered observations of a random sample of size 3 from a distribution with density

f(x) = \begin{cases} 1, & \theta - \frac{1}{2} < x < \theta + \frac{1}{2}, \\ 0, & \text{elsewhere.} \end{cases}

For $\alpha = \beta = 0.4$ show that $\mathbb {P}\left(\theta - \alpha < y_{2} < \theta + \beta \right) = 0.944$ .

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Problem 243

Let $Y_{1} < Y_{2} < Y_{3}$ be ordered observations of a random sample of size 3 from a distribution with density

f(x) = \begin{cases} 2x; & 0 < x < 1, \\ 0, & \text{elsewhere.} \end{cases}

Show that $Z_{1} = Y_{1}/Y_{2}, Z_{2} = Y_{2}/Y_{3}$ , and $Z_{3} = Y_{3}$ are completely independent.

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Problem 244

If $X_{1}, X_{2}$ are independent $\beta$ variables with densities $\beta (n_{1}, n_{2})$ and $\beta (n_{1}+1/2, n_{2})$ , respectively, where for $n_{1} > 0, n_{2} > 0$ ,

\beta (m, n) = \frac{1}{B(m,n)} x^{m-1} (1-x)^{n-1}, \qquad 0 < x < 1.

Show that $Y = \sqrt{X_{1} X_{2}}$ has also the $\beta$ distribution $\beta (2n_{1}, 2n_{2})$ .

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Problem 245

The random pair $(X, Y)$ has density

f(x,y) = \frac{1}{\Gamma (m)\Gamma (n)} x^{m-1} (y-x)^{n-1} e^{-y}, \qquad 0 < x < y < \infty .

Find the distribution of $Z = Y - X$ . Are $Z$ and $X$ independent? (cf. Exercise 174).

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Problem 246

Show that $Z = 2\sqrt{XY}$ , where $X$ and $Y$ are two independent $\Gamma$ variables with parameters $(\lambda_{1}, n)$ and $(\lambda_{2}, n+1/2)$ , respectively, has the $\Gamma$ distribution with parameters $(\lambda , 2n)$ where $\lambda = \sqrt{\lambda_{1}\lambda_{2}}$ .

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Problem 247

For the pair $(X, Y)$ of Exercise 245 show that $W = X/Y$ has the density $\beta (m, n)$ of Exercise 244.

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Problem 248

If $X$ and $Y$ are independent $\mathscr {N}\left(0, \sigma^{2}\right)$ , show that

Z = \frac{XY}{\sqrt{X^{2}+Y^{2}}}, \qquad W = \frac{X^{2}-Y^{2}}{\sqrt{X^{2}+Y^{2}}},

are also independent normal.

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Problem 249

Let $X$ have an $n$ -variate normal distribution $\mathscr {N}\left(\mu , \Sigma \right)$ . Find the density of $Y = AX$ where $A$ is a nonsingular matrix.

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Problem 250

The points of impact of two players $A_{1}, A_{2}$ on a vertical target follow the circular normal distribution with mean the center of the target and variances $\sigma_{1}^{2}$ and $\sigma_{2}^{2}$ , respectively. Find the probability that the impact point of $A_{1}$ is closer to the center than that of $A_{2}$ .

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