Math Problems Vault

Problem 251

If the independent random variables $X_{1}, \ldots , X_{n}, \ldots$ satisfy the condition

\operatorname {Var}\left[X_{i}\right] \leq c < \infty , \qquad i = 1, 2, \ldots ,

then the SLLN holds.

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Problem 252

In a sequence of random variables $X_{1}, \ldots , X_{n}, \ldots$ suppose that $X_{k}$ depends only on $X_{k-1}, X_{k+1}$ , but that it is independent of all the other random variables ( $k = 2, 3, \ldots$ ). Show that if $\operatorname {Var}\left[X_{i}\right] \leq N < \infty$ ( $i = 1, 2, \ldots$ ), then the WLLN holds.

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Problem 253

If for every $n$ , $\operatorname {Var}\left[X_{i}\right] \leq c < \infty$ and $\operatorname {Cov}\left[ X_{i}, X_{j} \right] < 0$ ( $i, j = 1, 2, \ldots , n$ ), then the WLLN holds.

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Problem 254

Let $\left\{ X_{n}\right\}$ be a sequence of random variables so that $\operatorname {Var}\left[X_{i}\right] \leq c < \infty$ ( $i = 1, 2, \ldots$ ) and $\operatorname {Cov}\left[ X_{i}, X_{j} \right] \to 0$ when $\left|i - j\right| \to \infty$ then the WLLN holds (Theorem of Bernstein).

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Problem 255

(a)(b)(c)

If $X_{n} \xrightarrow {L} X$ , $Y_{n} \xrightarrow {P} c$ , then

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(a)

$X_{n} + Y_{n} \xrightarrow {L} X + c$ ;

(b)

$X_{n} Y_{n} \xrightarrow {L} cX$ ;

(c)

$X_{n} / Y_{n} \xrightarrow {L} X / c$ .

Problem 256

Let $\xi_{n} = (X_{1n}, X_{2n}, \ldots , X_{kn})'$ ( $n = 1, 2, \ldots$ ) be a sequence of random vectors and $\xi = (X_{1}, X_{2}, \ldots , X_{k})'$ a random vector with distribution function $F(x_{1}, \ldots , x_{k})$ . If

\sum _{i=1}^{k} c_{i} X_{in} \xrightarrow {L} \sum _{i=1}^{k} c_{i} X_{i}

for all constants $c_{1}, \ldots , c_{k}$ , then the joint distribution function $F_{\xi_{n}}$ of $X_{1n}, \ldots , X_{kn}$ has a limit and

F_{\xi _{n}}(x_{1}, \ldots , x_{k}) \to F(x_{1}, \ldots , x_{k}).

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Problem 257

Let $\left\{ X_{n}\right\}$ be a sequence of independent random variables uniformly bounded, that is, there is a constant $M$ such that $\mathbb {P}\left(X_{k} \leq M\right) = 1$ for all $k = 1, 2, \ldots$ , and suppose that $\operatorname {Var}\left[X_{k}\right] \neq 0$ for every $k$ . If $s_{n}^{2} \to \infty$ , then the CLT holds (cf. (8.13)).

Formula (8.13), Lyapunov's CLT: let $X_{1}, \ldots , X_{n}, \ldots$ be a sequence of independent random variables with $\mathbb {E}\left[X_{i}\right] = \mu_{i}$ , $\operatorname {Var}\left[X_{i}\right] = \sigma_{i}^{2}$ , $\beta_{i} = \mathbb {E}\left[\left|X_{i} - \mu_{i}\right|^{3}\right]$ , and set $s_{n} = \left(\sum_{i=1}^{n} \sigma_{i}^{2}\right)^{1/2}$ , $B_{n} = \left(\sum_{i=1}^{n} \beta_{i}\right)^{1/3}$ ; if $B_{n}/s_{n} \to 0$ as $n \to \infty$ , then

S_{n}^{*} = \frac{\sum _{i=1}^{n} (X_{i} - \mu _{i})}{s_{n}} \xrightarrow [n \to \infty ]{L} N(0, 1).

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Problem 258

Show that

\lim _{n \to \infty } e^{-n} \sum _{k=0}^{n} \frac{n^{k}}{k!} = \frac{1}{2}.

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Problem 259

If $X$ and $Y$ are independent Poisson variables with parameters $\lambda_{1}$ and $\lambda_{2}$ , respectively, show that

\frac{X - Y - (\lambda _{1} - \lambda _{2})}{(X+Y)^{1/2}} \xrightarrow [\substack {\lambda _{1} \to \infty \\ \lambda _{2} \to \infty }]{L} N(0, 1).

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Problem 260

(a)(b)

Show that for the sequences $\left\{ X_{n}\right\}$ of independent random variables with

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(a)

$\displaystyle \mathbb {P}\left(X_{n} = \pm 1\right) = \frac{1 - 2^{-n}}{2}, \qquad \mathbb {P}\left(X_{n} = \pm 2^{n}\right) = \frac{1}{2^{n+1}}, \qquad n = 1, 2, \ldots ,$

(b)

$\mathbb {P}\left(X_{n} = \pm n^{\lambda }\right) = 1/2$ ,

the CLT holds.

Problem 261

Compare the results given by Chebyshev's inequality and the CLT for the probabilities

\mathbb {P}\left(-k < S_{n}^{*} \leq k\right) \qquad \text{for} \qquad k = 1, 2, 3.

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Problem 262

A variable $X$ has the Pareto distribution with density

f(x) = 24 x^{-4}, \qquad x \geq 2.

Give the graph of the function $g(\delta ) = \mathbb {P}\left(\left|X - \mu \right| > \delta \right)$ where $\mu = \mathbb {E}\left[X\right]$ , and compare this with the upper bound given by Chebyshev's inequality.

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Problem 263

(a)(b)

Given that $\operatorname {Var}\left[X\right] = 9$ , find the number $n$ of observations (the sample size) required in order that with probability less than $5\%$ the mean of the sample differs from the unknown mean $\mu$ of $X$ more,

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(a)

than $5\%$ of the standard deviation of $X$ ,

(b)

than $5\%$ of $\mu$ given that $\mu > 5$ .

Compare the answers obtained by using Chebyshev's inequality and the CLT.

Problem 264

In a poll designed to estimate the percentage $p$ of men who support a certain bill, how many men should be questioned in order that, with probability at least $95\%$ , the percentage of the sample differs from $p$ less,

than $1\%$ ,
than $5\%$ , given that,
$p < 30\%$ , and
$p$ is completely unknown.

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Problem 265

Each of the 300 workers of a factory takes his lunch in one of three competing restaurants. How many seats should each restaurant have so that, on average, at most one in 20 customers will remain unseated?

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Problem 266

The round-off error to the second decimal place has the uniform distribution on the interval $(-0.05, 0.05)$ . What is the probability that the absolute error in the sum of 1,000 numbers is less than 2?

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Problem 267

(a)(b)

The daily income of a card player has uniform distribution in the interval $(-40, 50)$ .

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(a)

What is the probability that he wins more than 500 drachmas in 60 days?

(b)

What amount $c$ (gain or loss) is such that at most once in 10 times on average the player will have income less than $c$ during 60 days?

Problem 268

In the game of roulette the probability of winning is $18/37$ . Suppose that in each game a player earns 1 drachma or loses 1 drachma. How many games must be played daily in a casino in order that the casino earns, with probability $1/2$ , at least 1,000 drachmas daily? Then find the percentage of days on which the casino has a loss.

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Problem 269

Show that the $\Gamma$ distribution of Exercise 202 tends to the normal distribution as $s \to \infty$ .

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