Using a standard uniform random variable,[0,1], we want to generate samples of a random variable, X, whose probability density function (pdf), p, (x), is sketched below. Px(x) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 02 0.1 0 -3 -2.5 -2 -1.5 x -1 -0.5 0 andom a) Write down the mathematical expression of the above pdf. Check whether it is a valid pdf. b) Find the required transformation equation. c) Using MATLAB or a similar computer program, generate 100 samples of X and sketch their histogram. d) On the histogram, sketch the true pdf. e) Repeat parts c) and d) for 1000 and 10000 samples of X. f) Comment on your results. g) In your report, include your software codes with explanations of steps.

A random variable X has the following probability mass function for x = 0, 1, 2, 5. Based (PMF): P(X= x) = (x+1) 12 on this PMF, what is P(X≥ 2)? 5/12 1/2 3/4 11/12

part 1 of 2 Consider a discrete random variable X that is defined by its piece-wise defined cumulative distribution function: Pr(X < x) 0.8- 0.6(1, 0.5) 0.4 0.2 (4,1) (3, 0.9) -2 -1 0 1 2 3 4 5 6 What is the probability distribution of X? part 2 of 2 Compute the expected value (mean) for X. 1. 2. 3. 4. x 1 3 4 x 1 3 4 x 1 3 4 x 1 3 4 Pr(X = x) 0 0.1 0.4 Pr(X = x) 0.5 0.9 1 Pr(X = x) 0.5 0.4 0.1 Pr(X = x) 10 50 40 1. At least 5.0 2. At least 2.0, but less than 3.0 3. At least 4.0, but less than 5.0 4. At least 3.0, but less than 4.0 5. Less than 2.0

Two independent random variables are defined with the following distributions: Xp Y p 1 0.5 1 0.2 2 0.2 2 0.2 3 0.3 4 0.6 (a) Determine the PDF of the sum of X and Y using convolutions. (b) Determine the mean and variance of the sum by treating the result as the sum of two independent random variables.