# Exam 2_SOLUTION

Problems1. (24 points total, 4 points each) True/False Questions. Circle either True or False.
(a) True or False: A probability density function fX(x) for a continuous random variable X can take values greater than 1.
(b) True or False: A cumulative distribution function (CDF) FZ(z) can take values greater than 1.
(c) True or False: The integral of the probability density function fX(x), over −∞ < x < ∞, is always equal to 1.
(d) True or False: The probability that X = 1, for a discrete random variable X, is the probability mass function fX(x), evaluated at x = 1.
(e) True or False: EX2is equal to (E [X])2 for any continuous random variable X.
(f) True or False: For a Binomial random variable K, which represents the number of successes in n trials where each trial is a success with probability p, the expected value E [K] is equal to np.

2. (6 points) Circle the one correct answer: Let X be a random variable with standard deviation σX and mean µX. What is NOT a valid expression for the variance of X? (a) EX2 (b) (σX)2 (c) Eh(X − µX)2i (d) EX2− (µX)2
3. (6 points) Circle the one correct answer: We ﬂip a fair coin 10 times. What is the probability of getting 5 tails? (a) 10 5 1 210 (b) 10! 5!  1 210 (c) 0.5 (d) 0.1 (e) 1 210
4. (6 points) Circle the one correct answer: For continuous random variables X and Y , we ﬁnd that E [X] = 1.0, E [Y ] = 4.0, E [XY ] = 5.0, EX2= 1.5, and EY 2= 20.0. What is the correlation coeﬃcient ρXY ?
(a) 0.354 (b) 0.5 (c) 0.707 (d) 1.0 (e) 2.0

5. (16 points total) Random variable X is Gaussian with mean 2.0 and standard deviation 4.0.
(a) (7 points) Find P [X 7.5].
(b) (9 points) Find a value d such that P [2.0 − d < X < 2 + d] = 0.90.
ECE 3530 / CS 3130 Spring 2014 5
6. (14 points) Let X have the pdf:
f(x) = 3 x4 , x 1 0, o.w.
(a) Find Var [X].