Examples of characteristic functions

If $\phi$ is a characteristic function, show that $\bar{\phi }, \phi^{2},\left|\phi \right|^{2}, \operatorname {Re}(\phi )$ are characteristic functions. Show that $\left|\phi \right|$ is not necessarily a characteristic function.

Show that

where $M_{X}$ is the moment generating function of $X$. Deduce that, if $X$ has the $N(0,1)$ distribution,

Let $X$ have the $\Gamma (\lambda , m)$ distribution and let $Y$ be independent of $X$ with the beta distribution with parameters $n$ and $m-n$, where $m$ and $n$ are non-negative integers satisfying $n \leq m$. Show that $Z = X Y$ has the $\Gamma (\lambda , n)$ distribution.

Find the characteristic function of $X^{2}$ when $X$ has the $N\left(\mu , \sigma^{2}\right)$ distribution.

Let $X_{1}, X_{2}, \ldots$ be independent $N(0,1)$ variables. Use characteristic functions to find the distribution of:

Let $X_{1}, X_{2}, \ldots , X_{n}$ be such that, for all $a_{1}, a_{2}, \ldots , a_{n} \in \mathbb {R}$, the linear combination $a_{1} X_{1}+ a_{2} X_{2}+\cdots +a_{n} X_{n}$ has a normal distribution. Show that the joint characteristic function of the $X_{m}$ is $\exp \left(i \mathbf{t} \mathbf{\mu }^{\prime }-\frac{1}{2} \mathbf{t} \mathbf{\mathbf{ t }^{ \prime }}\right)$, for an appropriate vector $\mathbf{\mu }$ and matrix $\mathbf{V}$. Deduce that the vector $\left(X_{1}, X_{2}, \ldots , X_{n}\right)$ has a multivariate normal density function so long as $\mathbf{V}$ is invertible.

Let $X$ and $Y$ be independent $N(0,1)$ variables, and let $U$ and $V$ be independent of $X$ and $Y$. Show that $Z = (U X+V Y) / \sqrt{U^{2}+V^{2}}$ has the $N(0,1)$ distribution. Formulate an extension of this result to cover the case when $X$ and $Y$ have a bivariate normal distribution with zero means, unit variances, and correlation $\rho$.

Let $X$ be exponentially distributed with parameter $\lambda$. Show by elementary integration that $\mathbb {E}\left[e^{i t X}\right] = \lambda /(\lambda -i t)$.

Find the characteristic functions of the following density functions:

Is it possible for $X, Y$, and $Z$ to have the same distribution and satisfy $X = U(Y+Z)$, where $U$ is uniform on $[0,1]$, and $Y, Z$ are independent of $U$ and of one another? (This question arises in modelling energy redistribution among physical particles.)

Find the joint characteristic function of two random variables having a bivariate normal distribution with zero means. (No integration is needed.)

Sampling from the normal distribution. Let $X_{1}, X_{2}, \ldots , X_{n}$ be independent random variables with the $N\left(\mu , \sigma^{2}\right)$ distribution, where $\sigma > 0$. Let

be the sample mean and variance. Show that $\operatorname {Cov}\left[\bar{X}, X_{i}-\bar{X}\right] = 0$, and deduce that $\bar{X}$ and $S^{2}$ are independent.

Show that

and use characteristic functions to prove that $(n-1) S^{2} / \sigma^{2}$ has the $\chi^{2}(n-1)$ distribution.

Let the vector $\left(X_{1}, X_{2}, \ldots , X_{n}\right)$ have a multivariate normal distribution with zero means. Show that, for $n$ odd, $\mathbb {E}\left[X_{1} X_{2} \cdots X_{n}\right] = 0$, while for $n = 2 m$,

that is, the sum over all products of expectations of $m$ distinct pairs of variables.

Let $X_{1}, X_{2}, \ldots , X_{n}$ be independent random variables with, respectively, the exponential distribution with parameter $\lambda_{r}$ for $r = 1,2, \ldots , n$.

Use moment generating functions to find the density function of the sum $S = X_{1}+X_{2}+\cdots +X_{n}$. Deduce that, for any set $\left\{ \lambda_{r}: 1 \leq r \leq n\right\}$ of distinct positive numbers,

Let $X$ have the $N(0,1)$ distribution, and let $f: \mathbb {R} \rightarrow \mathbb {R}$ be sufficiently smooth. Show that

and deduce that $\mathbb {E}\left[X f(X)\right] = \mathbb {E}\left[f^{\prime }(X)\right]$.

Find the characteristic function of the $N(0,1)$ distribution without using the methods of complex analysis. [Hint: Consider the derivative of the characteristic function, and use an appropriate theorem to differentiate through the integral.]

Let $X$ be a non-negative random variable with $0 < \mu = \mathbb {E}\left[X\right] < \infty$. The random variable $Y$ is said to have the size-biased $X$ distribution if $d F_{Y}(x) = (x / \mu ) d F_{X}(x)$ for all $x$. (You may think of this as saying that $f_{Y}(x) \propto x f_{X}(x)$ if $X$ is either discrete with mass function $f_{X}$, or continuous with density function $f_{X}$.)

Show that: