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?

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**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.

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**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.

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