Math Problems Vault

Problem 11.1

Find the characteristic function of a random variable $X$ such that $\mathbb {P}\left(\{ X=1\} \right)=\mathbb {P}\left(\{ X=-1\} \right)=\frac{1}{2}$ .

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Problem 11.2

Find the characteristic function of a random variable $X$ such that $\mathbb {P}\left(\{ X=1\} \right)=p$ and $\mathbb {P}\left(\{ X=0\} \right)=q$ , where $p+q=1$ (Bernoulli distribution).

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Problem 11.3

Find the characteristic function of a random variable $X$ such that $\mathbb {P}\left(\{ X=k\} \right)=2^{-k}, k=1,2, \ldots$ .

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Problem 11.4

Find the characteristic function of a random variable $X$ with Poisson distribution. Can you sketch the graph of $\phi_{X}(t)$ ?

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Problem 11.5

Find the characteristic function of a random variable $X$ with uniform distribution over the interval $[-1,1]$ . Sketch the graph of this characteristic function.

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Problem 11.6

Find the characteristic function $\phi_{X}(t)$ of a random variable $X$ having distribution with density

f_{X}(x)= \begin{cases} 1-\left|x\right| & \text{ if }\left|x\right| \leqslant 1 \\ 0 & \text{ otherwise } \end{cases}

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Problem 11.7

Find the characteristic function $\phi_{X}(t)$ of a random variable $X$ with distribution

P_{X}(A)=\frac{1}{2} \; \mathbb {1}_{A}(0)+\frac{1}{4} \int _{-1}^{1} \; \mathbb {1}_{A}(x) \mathrm{d} x

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Problem 11.8

What is the characteristic function $\phi_{X}(t)$ of a random variable $X$ with exponential distribution? Sketch the graph of $\phi_{X}(t)$ .

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Problem 11.9

What is the characteristic function $\phi_{X}(t)$ of a normally distributed random variable $X$ with mean 0 and variance 1 ?

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Problem 11.10

(a)(b)(c)

Show that for any random variable $X$

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(a)

$\phi_{X}(0)=1$ ,

(b)

$\left|\phi_{X}(t)\right| \leqslant 1$ for all $t \in \mathbb {R}$ .

(c)

$\phi_{X}(-t)=\overline{\phi_{X}(t)}$ for all $t \in \mathbb {R}$ . (By $\bar{z}$ we denote the complex conjugate of $z \in \mathbb {C}$ ; that is, if $z=a+i b$ , where $a, b \in \mathbb {R}$ , then $\bar{z}=a-i b$ .)

Problem 11.11

Let $X$ and $Y$ be random variables such that $Y=a X+b$ , where $a, b \in \mathbb {R}$ . Show that

\phi _{Y}(t)=e^{i t b} \phi _{X}(a t)

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Problem 11.12

Use the formula in Problem 11.11 and the characteristic function of the normal distribution with mean 0 and variance 1 obtained in Problem 11.9 to find the characteristic function of the normal distribution with arbitrary mean $m$ and variance $\sigma^{2}>0$ .

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Problem 11.13

Let $X$ be a random variable such that $\mathbb {P}\left(\{ X \in \mathbb {Z}\} \right)=1$ . Demonstrate that $\phi_{X}(t)$ is a periodic function with period $2 \pi$ ; that is, $\phi_{X}(t+2 \pi )=\phi_{X}(t)$ for all $t$ .

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Problem 11.14

Is it true that if $\phi_{X}(2 \pi )=1$ , then $\mathbb {P}\left(\{ X \in \mathbb {Z}\} \right)=1$ ?

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Problem 11.15

(a)(b)(c)

Let $X$ be a random variable and let $a \neq 0$ . Show that the following conditions are equivalent:

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(a)

$\phi_{X}(a)=1$ ;

(b)

$\phi_{X}$ is periodic with period $\left|a\right|$ ;

(c)

$P\left(\left\{ \frac{a}{2 \pi } X \in \mathbb {Z}\right\} \right)=1$ .

Problem 11.16

If $X$ is a random variable with density, then $\phi_{X}(t)<1$ for all $t \neq 0$ .

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Problem 11.17

Let $X$ be a random variable taking values in a finite set. Verify that the moments $\mathbb {E}\left[X^{n}\right]$ of $X$ are given by

\mathbb {E}\left[X^{n}\right]=\frac{1}{i^{n}} \frac{d^{n}}{d t^{n}} \phi _{X}(0)

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Problem 11.18

Use the formula in Theorem 11.1 to obtain an expression for the variance of $X$ in terms of the characteristic function $\phi_{X}$ and its derivatives at zero.

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Problem 11.19

Use Theorem 11.1 to compute the expectation $\mathbb {E}\left[X\right]$ and the second moment $\mathbb {E}\left[X^{2}\right]$ of a random variable $X$ with Poisson distribution.

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Problem 11.20

Use Theorem 11.1 to compute the expectation $\mathbb {E}\left[X\right]$ and the second moment $\mathbb {E}\left[X^{2}\right]$ of a random variable $X$ with normal distribution.

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Problem 11.21

Let $X$ be a random variable such that $\mathbb {P}\left(\{ X \in \mathbb {Z}\} \right)=1$ . Show that for all $n \in \mathbb {Z}$ ,

\mathbb {P}\left(\{ X=n\} \right)=\frac{1}{2 \pi } \int _{0}^{2 \pi } e^{-i t n} \phi _{X}(t) \mathrm{d} t

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Problem 11.22

What is the distribution of a random variable $X$ with characteristic function

\phi _{X}(t)=\frac{1}{2} e^{-i t}+\frac{1}{3}+\frac{1}{6} e^{2 i t} ?

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Problem 11.23

Find the distribution of a random variable $X$ with characteristic function

\phi _{X}(t)=\cos \left(\frac{t}{2}\right)

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Problem 11.24

Find the distribution of a random variable $X$ with characteristic function

\phi _{X}(t)=\frac{2}{3 e^{i t}-1}

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Problem 11.25

Find the distribution of a random variable with characteristic function $e^{-\left|t\right|}$ .

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Problem 11.26

What is the distribution of a random variable with characteristic function $e^{-t^{2} / 2}$ ?

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Problem 11.27

Find the distribution of a random variable with characteristic function $\frac{1}{1+t^{2}}$ .

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Problem 11.28

Find the distribution of a random variable with characteristic function $\frac{1}{3} e^{i t \sqrt{2}}+\frac{2}{3} e^{i t \sqrt{3}}$ .

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Problem 11.29

What is the distribution of a random variable with characteristic function $\frac{1}{2} e^{-t^{2} / 2}+\frac{1}{2}$ ?

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Problem 11.30

Prove Theorem 11.4.

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Problem 11.31

Use Theorem 11.4 to find the characteristic function of a random variable with the binomial distribution.

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Problem 11.32

Let $X$ and $Y$ be independent random variables with Poisson distribution. Verify that $X+Y$ is also a random variable with Poisson distribution. If $\lambda$ and $\mu$ are the parameters of $X$ and $Y$ , respectively, what is the parameter of $X+Y$ ?

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Problem 11.33

Let $X$ and $Y$ be normally distributed independent random variables. Show that $X+Y$ is also normally distributed. If $X$ has mean $m$ and variance $\sigma^{2}$ and $Y$ has mean $\tilde{m}$ and variance $\tilde{\sigma }^{2}$ , what is the mean and variance of $X+Y$ ?

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