ECE-QE CS1-2012 - Rhea

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?

Click here to view student answers and discussions

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.

Click here to view student answers and discussions

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.

Click here to view student answers and discussions

Back to ECE Qualifying Exams (QE) page