Sabancı University MATH 203, Homework 11, Spring 2018 1. With reference to Exercise 3.42, where the joint probability distribution of X and Y is as shown in the table below, Y 0 X 1 2 0 1 12 1 6 1 24 1 1 4 1 4 1 40 2 1 8 1 20 3 1 120 (a) find the conditional expectation of X given Y = 1; (b) find the conditional expectation of Y 2 given X = 0. 2. Let X and Y be two continuous random variables with the joint probability density ( 2x(x − y) for 0 < x < 1, −x < y < x f (x, y) = 0 elsewhere Find the conditional expected value of |Y | given that X = 12 . 3. If X and Y are independent, determine Cov(T, S), where T and S are defined by T = X + 2Y + 3 and 2 = 1 and σ 2 = 1 . S = −2X + 4Y . It is given that that σX Y 4 4. Let X1 and X2 be random variables corresponding to scores on the Math 203 exam 1 and exam 2. History suggests that X1 and X2 are independent and normally distributed with means of 60 and 40, standard deviations of 15 and 20, respectively. If the average passing grade from this course is 40, (a) what is the probability that a randomly selected student will pass the course. (b) what is the probability that a student will pass this course given that his or her grade from the first exam is 25. 5. Two independent random variables X1 and X2 have the same moment generating function 2 M (t) = et+t for t ∈ R. Calculate the moment generating function of the random variable S = 2X1 + X2 + 1. 6. Let X be a discrete random variable taking values x = 1, 2, 3 according to the probability distribution 3 fX (1) = , 6 fX (2) = 1 6 fX (3) = 2 6 Find the probability distribution fY (y) of the random variable Y = X/(1 + X). 7. Exercise 7.3: If X has the uniform density with the parameters α =√ 0 and β = 1, use the distribution function technique to find the probability density of the random variable Y = X.