Math Problems Vault

Problem 14.1

Let $g(x)$ be a nonnegative increasing function and $\xi$ an arbitrary random variable. Prove that for any $x$ the inequality

\mathbb {P}\left\{ \xi \geqslant x\right\} \leqslant \frac{\mathbb {E}\left[g(\xi )\right]}{g(x)}

holds.

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Problem 14.2

Let $\mathbb {E}\left[\xi^2\right] < \infty$ . Prove that

\mathbb {P}\left\{ \left|\xi - \mathbb {E}\left[\xi \right]\right| > 3\sqrt{\operatorname {Var}\left[\xi \right]}\right\} \leqslant 1/9.

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Problem 14.3

Prove that for arbitrary random variables $\xi$ and $\eta$

\mathbb {E}\left[\left|\xi \eta \right|\right] \leqslant \sqrt{\mathbb {E}\left[\xi ^2\right]\mathbb {E}\left[\eta ^2\right]}.

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Problem 14.4

(a)(b)

Let $\mathbb {E}\left[\left|\xi \right|^r\right] < \infty$ . Prove that:

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(a)

$(\mathbb {E}\left[\left|\xi \right|^s\right])^r \leqslant (\mathbb {E}\left[\left|\xi \right|^r\right])^s$ for any $s \in (0, r)$ ;

(b)

$(\mathbb {E}\left[\left|\xi \right|^s\right])^{1/s} \leqslant (\mathbb {E}\left[\left|\xi \right|^r\right])^{1/r}$ for any $s \in (0, r)$ .

Problem 14.5

Prove that $\mathbb {E}\left[\left|\xi \right|\right] \leqslant \sqrt[r]{\mathbb {E}\left[\left|\xi \right|^r\right]}$ for $r > 1$ .

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Problem 14.6

Prove that for a random variable $\xi$ with finite variance and any $x > 0$

\mathbb {P}\left\{ \left|\xi - \mathbb {E}\left[\xi \right]\right| > x\right\} \leqslant \frac{2\operatorname {Var}\left[\xi \right]}{x^2 + \operatorname {Var}\left[\xi \right]}.

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Problem 14.7

Let $\xi$ be a random variable with $\mathbb {E}\left[e^{h\xi }\right] < \infty$ , $h > 0$ . Prove that for any $x$

\mathbb {P}\left\{ \xi \geqslant x\right\} \leqslant \mathbb {E}\left[e^{h\xi }\right]/e^{hx}.

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Problem 14.8

(a)(b)(c)

Let the random variable $\xi$ have a Poisson distribution with parameter $\lambda$ . Prove that for any $x > 0$ :

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(a)

$\mathbb {P}\left\{ \xi \geqslant x\right\} \leqslant \lambda /x$ ;

(b)

$\mathbb {P}\left\{ \xi \geqslant x\right\} \leqslant e^\lambda /2^x$ ;

(c)

$\mathbb {P}\left\{ \xi \geqslant 2\lambda \right\} \leqslant 1/\max (2, \lambda )$ .

Problem 14.9

(a)(b)(c)

Let the random variable $\xi$ have:

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(a)

a Poisson distribution with parameter $\lambda$ ;

(b)

an exponential distribution with parameter $\alpha$ ;

(c)

the standard normal distribution.

Determine the value of the parameter $h$ minimizing the expression on the right-hand side of the inequality in problem 14.7. Find the value of this minimum.

Problem 14.10

(a)(b)

Prove the following inequalities:

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(a)

$\mathbb {E}\left[\xi^4\right] \geqslant (\mathbb {E}\left[\xi \right])^4$ ;

(b)

$\mathbb {E}\left[\xi^2\right]\mathbb {E}\left[\xi^4\right] \geqslant (\mathbb {E}\left[\xi^3\right])^2$ .

When can equality hold in these inequalities?

Problem 14.11

Is the following reasoning valid: «It is 1 km from home to work, and I walk at an average speed of 5 km/h, so on average my commute takes 12 minutes»?

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Problem 14.12

Prove that if the random variable $\xi$ has a moment of order $k$ , then its distribution function $F$ satisfies the relation

\lim _{x \to \infty } x^k (1 - F(x) + F(-x)) = 0.

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Problem 14.13

Prove that $\mathbb {E}\left[\xi^2\right] = 0$ is equivalent to $\mathbb {P}\left\{ \xi = 0\right\} = 1$ .

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Problem 14.14

Suppose $\mathbb {E}\left[\left|\xi \right|^p\right] = 0$ for some $p > 0$ . What can be said about the random variable $\xi$ ?

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Problem 14.15

Let the random variable $\xi$ have finite expectation, with $\mathbb {E}\left[\xi \; \mathbb {1}_{A}\right] = 0$ for every event $A$ . Prove that $\mathbb {P}\left\{ \xi = 0\right\} = 1$ .

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Problem 14.16

Prove that for any nonnegative random variable $\xi$ with finite variance, the inequality

\mathbb {P}\left\{ \xi = 0\right\} \leqslant \frac{\operatorname {Var}\left[\xi \right]}{\operatorname {Var}\left[\xi \right] + (\mathbb {E}\left[\xi \right])^2}

holds.

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