Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

August 2012

## Question

Problem 1. 25 pts Consider a random experiment in which a point is selected at random from the unit square (sample space $\mathcal{S} = [0,1] \times [0,1]$). Assume that all points in $\mathcal{S}$ are equally likely to be selected. Let the random variable $\mathbf{X}(\omega)$ be the distance from the outcome $\omega$ to the nearest edge (i.e. the nearest point on one of the four sides) of the unit square.

(a) Find the c.d.f. of $\mathbf{X}$. Draw a graph of the c.d.f..

(b) Find the p.d.f. of $\mathbf{X}$. Draw a graph of the p.d.f..

(c) What is the probability that $\mathbf{X}$ is less than 1/8?

Problem 2. 25 pts

State and prove the Chebyshev inequality for random variable $\mathbf{X}$ with mean $\mu$ and variance $\sigma^2$. In constructing your proof, keep in mind that $\mathbf{X}$ may be either a discrete or continuous random variable.

Problem 3. 25 pts

Let $\mathbf{X}_{1} \dots \mathbf{X}_{n} \dots$ be a sequence of independent, identical distributed random variables, each uniformly distributed on the interval [0, 1], an hence having pdf
$f_{X}\left(x\right)=\begin{cases} \begin{array}{lll} 1, \text{ for } 0 \leq x \leq1\\ 0, \text{ elsewhere. } \end{array}\end{cases}$

Let $\mathbf{Y}_{n}$ be a new random variable defined by

$\mathbf{Y}_{n} = min \,\{{ \mathbf{X}_1, \mathbf{X}_2, \dots \mathbf{X}_n} \}$


(a) Find the pdf of $\mathbf{Y}_{n}$.

(b) Does the sequence $\mathbf{Y}_{n}$ converge in probability?

(c) Does the sequence $\mathbf{Y}_{n}$ converge in distribution? If yes, specify the cumulative function of the random variable it converges to.