Problem set 3: Characteristic Functions

Find the characteristic function of a random variable \(\xi\) having the following probability distribution:

  1. \(\operatorname{Bin}(n,p)\);

  2. \(\mathcal{N}(a,\sigma^2)\);

  3. \(\operatorname{Gamma}(\alpha,\lambda)\).

Let \(\phi(t)\) and \(\varphi(t)\) be some characteristic functions. Prove that

\[ \frac{\phi(t)+\varphi(t)}{2}, \qquad \phi(t)\varphi(t), \qquad |\phi(t)|^2 \]

are also characteristic functions of some random variables.

For which non-negative integers \(n\) does it hold that

\[ \phi(t)=e^{-|t|^n} \]

is a characteristic function?

Which of the following functions represent the characteristic function of some probability distribution?

\[ \sin t; \qquad 1+\sin t; \qquad \cos t; \]

\[ e^{-|t|}\mathbf{1}_{\{t<0\}} + \frac{1}{1+t^2}\mathbf{1}_{\{t\geq 0\}}; \qquad \frac{1}{1+t^2}. \]

Let \(\xi_1,\xi_2\) be independent random variables. Using characteristic functions, find the probability distribution of \(\xi_1+\xi_2\), where

  1. \(\xi_i\sim\mathcal{N}(a_i,\sigma_i^2)\);

  2. \(\xi_i\sim\operatorname{Gamma}(\alpha_i,\lambda)\);

  3. \(\xi_i\) has a Cauchy distribution with parameter \(\theta_i\).

Let \(\xi\) and \(\eta\) be independent random variables with standard normal distribution.

Find the characteristic function of

\[ \xi\eta. \]

Compute

\[ \mathbb{E}\xi^k \]

using characteristic functions. Consider

\[ \xi\sim\operatorname{Exp}(\lambda). \]

Find the characteristic function of a random variable \(\xi\) having the following probability distribution:

  1. \(\operatorname{Pois}(\lambda)\);

  2. \(\operatorname{Geom}(p)\);

  3. Laplace with parameter \(\theta>0\), that is,

\[ p_\xi(x)=\frac{1}{2\theta}e^{-|x|/\theta}; \]

  1. \(\operatorname{Cauchy}(\theta)\).

Which of the following functions represent the characteristic function of some probability distribution?

\[ \cos^2 t; \qquad \cos(t^2); \qquad \frac{1}{1+t^4}; \qquad \frac{1+3e^{it}+2e^{-it}}{6}. \]

Compute

\[ \mathbb{E}\xi^k \]

using characteristic functions. Consider

\[ \xi\sim\mathcal{N}(0,\sigma^2). \]

Let the random variable \(\xi\) have the following characteristic function:

\[ \phi(t)=(1-|t|)\mathbf{1}_{\{|t|\leq 1\}}. \]

Compute the density and expected value of \(\xi\).