Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

August 2007

## Question

1. (25 Points)

X and Y are iid random variable with

$P(X=i) = P(Y=i) = \frac {1}{2^i}\ ,i = 1,2,3,...$

a) Find $P(min(X,Y)=k)\$.

b) Find $P(X=Y)\$.

c) Find $P(Y>X)\$.

d) Find $P(Y=kX)\$.

2. (25 Points)

Let $\left\{ \mathbf{X}_{n}\right\} _{n\geq1}$ be a sequence of binomially distributed random variables, with the $n$ -th random variable $\mathbf{X}_{n}$ having pmf $p_{\mathbf{X}_{n}}\left(k\right)=P\left(\left\{ \mathbf{X}_{n}=k\right\} \right)=\left(\begin{array}{c} n\\ k \end{array}\right)p_{n}^{k}\left(1-p_{n}\right)^{n-k}\;,\qquad k=0,\cdots,n,\quad p_{n}\in\left(0,1\right).$

Show that, if the $p_{n}$ have the property that $np_{n}\rightarrow\lambda$ as $n\rightarrow\infty$ , where $\lambda$ is a positive constant, then the sequence $\left\{ \mathbf{X}_{n}\right\} _{n\geq1}$ converges in distribution to a Poisson random variable $\mathbf{X}$ with mean $\lambda$ .

Hint:

You may find the following fact useful:

$\lim_{n\rightarrow\infty}\left(1+\frac{x}{n}\right)^{n}=e^{x}.$

3. (25 Points)

Let $\mathbf{X}\left(t\right)$ be a real Gaussian random process with mean function $\mu\left(t\right)$ and autocovariance function $C_{\mathbf{XX}}\left(t_{1},t_{2}\right)$ .

(a)

Write the expression for the $n$ -th order characteristic function of $\mathbf{X}\left(t\right)$ in terms of $\mu\left(t\right)$ and $C_{\mathbf{XX}}\left(t_{1},t_{2}\right)$ .

ref.

There are the note about the n-th order characteristic function of Gaussians random process . The only difference between the note and this problem is that this problem use the $\mu\left(t\right)$ rather than $\eta_{\mathbf{X}}\left(t\right)=E\left[\mathbf{X}\left(t\right)\right]$ .

Solution

$\Phi_{\mathbf{X}\left(t_{1}\right)\cdots\mathbf{X}\left(t_{n}\right)}\left(\omega_{1},\cdots,\omega_{n}\right)=\exp\left\{ i\sum_{k=1}^{n}\mu_{\mathbf{X}}\left(t_{k}\right)\omega_{k}-\frac{1}{2}\sum_{j=1}^{n}\sum_{k=1}^{n}C_{\mathbf{XX}}\left(t_{j},t_{k}\right)\omega_{j}\omega_{k}\right\}$ .

(b)

Show that the probabilistic description of $\mathbf{X}\left(t\right)$ is completely characterized by $\mu\left(t\right)$ and autocovariance function $C_{\mathbf{XX}}\left(t_{1},t_{2}\right)$ .

4. (25 Points)

Let $\mathbf{X}_{1},\mathbf{X}_{2},\mathbf{X}_{3},\cdots$ be a sequence of independent, identically distributed random variables, each having Cauchy pdf $f\left(x\right)=\frac{1}{\pi\left(1+x^{2}\right)}\;,\qquad-\infty<x<\infty. Let \mathbf{Y}_{n}=\frac{1}{n}\sum_{i=1}^{n}\mathbf{X}_{i}.$ Find the pdf of $\mathbf{Y}_{n}$ . Describe how the pdf of $\mathbf{Y}_{n}$ depends on $n$ . Does the sequence $\mathbf{Y}_{1},\mathbf{Y}_{2},\mathbf{Y}_{3},\cdots$ converge in distribution? If yes, what is the distribution of the random variable it converges to?