Characteristic Functions

Find the characteristic function of the random variable $\xi$ if it has

1. a binomial distribution with parameters $(n, p)$;

2. a Poisson distribution with parameter $\lambda$;

3. a geometric distribution with parameter $p$;

4. a normal distribution with parameters $\left(a, \sigma^{2}\right)$;

5. a gamma distribution with parameters $(\alpha , \lambda )$;

6. a Laplace distribution with parameter $\theta >0$ (i.e. $p_{\xi }(x)=\frac{1}{2 \theta } e^{-\left|x\right| / \theta }$).

Using the inversion formula, find the characteristic function of the random variable $\xi$ if it has the Cauchy distribution with parameter $\theta$.

Let $\varphi (t)$ be a characteristic function. Show that the following inequalities hold:

1. $(\operatorname {Im} \varphi (t))^{2} \leq \frac{1}{2}(1-\operatorname {Re} \varphi (2 t))$

2. $(\operatorname {Re} \varphi (t))^{2} \leq \frac{1}{2}(1+\operatorname {Re} \varphi (2 t))$;

3. $\left|\frac{1}{2 h} \int_{t-h}^{t+h} \varphi (u) d u\right| \leq (1+\operatorname {Re} \varphi (h))^{\frac{1}{2}}$.

For which nonnegative integers $n$ is the function $\varphi (t)=e^{-|t|^{n}}$ a characteristic function?

Determine whether the following functions are characteristic functions:

1. $\cos t$

2. $\cos^{2} t$

3. $\cos t^{2}$

4. $e^{-\left|t\right|} I\left\{ t<0\right\} +\left(1+t^{2}\right)^{-1} I\left\{ t \geq 0\right\}$

5. $\frac{1}{1+t^{2}}$

6. $\frac{1}{1+t^{4}}$.

Let $\xi_{1}, \xi_{2}$ be independent random variables. Using characteristic functions, find the distribution of $\xi_{1}+\xi_{2}$ if

1. $\xi_{i}$ has a normal distribution with parameters $\left(a_{i}, \sigma_{i}^{2}\right)$

2. $\xi_{i}$ has a gamma distribution with parameters $\left(\alpha_{i}, \lambda \right)$,

3. $\xi_{i}$ has a Cauchy distribution with parameter $\theta_{i}$.

Let $\xi$ and $\eta$ be independent standard normal random variables. Find the characteristic function of the random variable $\xi \eta$.

Let $\left\{ \xi_{n}, n \in \mathbb {N}\right\}$ be a sequence of normal random variables. Prove that if $\xi_{n} \xrightarrow {d} \xi$, then $\xi$ is also a normal random variable or a constant.

A random variable $\xi$ takes only integer values, and $\varphi (t)$ is the characteristic function of $\xi$. Prove that for any natural $k$ we have

Using the characteristic function, find all moments of the random variable $\xi$ if

1. $\xi \sim \operatorname {Exp}(\lambda )$

2. $\xi \sim \mathcal{N}\left(0, \sigma^{2}\right)$