7 Math 161A - Spring 2017 1. Let X Binomial(n = 200, p = 0.7). Use the Normal approximation to the Binomial (and an appropriate continuity correction) to find P (122 X < 150). Show your work. You could use your Calculator to compute the above probability directly (without approximating anything) but that would not lead to full credit in this problem. 2. Two numbers are chosen (without replacement) from the set {1, 2, 3, 4}. Let X be the smaller number and Y the product of the two numbers. (a) Write down the probability mass function table for X and Y . (b) Find P (Y = 3X). (c) Without computing anything, do you think the correlation between X and Y will be positive or negative? Explain. 3. Let X and Y be discrete random variables with joint PMF table X Y 0 1 2 1 0.30 0.10 0.05 1 0.05 0.30 0.20 (a) Find the marginal distributions of X and Y and fill in the respective values in the table. (b) Find E(X) and E(Y ). (c) Find V (X) and V (Y ). (d) Find E(XY ). (e) Compute the covariance of X and Y . (f) Compute the correlation between X and Y . What does the sign of the correlation tell you about the joint behavior of X and Y ? (g) Find P (X = 2|Y = 1). (h) Find V (2Y 3X + 5). 4. Let Let X and Y be jointly distributed continuous random variables with joint PDF \u001a c( x)y 0 x 1, 0 y 2 f (x, y) = 0 otherwise (a) Find the value of c that makes this a valid joint PDF. (b) Compute P (0 X 0.5, 1 Y 1.2). (c) Find the marginal distributions fX (x) of X and fY (y) of Y . (d) Are X and Y independent? (e) Find the conditional PDF of X given that Y = 0.6. (f) Compute P (X > Y ). 1 Homework 7 Math 161A - Spring 2017 5. Let X and Y be independent Poisson random variables with rates 1 = 2 and 2 = 3, respectively. (a) Find the joint PMF of X and Y , e.g. find the function that stands for p(x, y) = P (X = x and Y = y). What are the possible values for x and y and what are the values of p(x, y)? Write the probabilities down as a function of x and y. (b) Find the probability that X + Y 1. (c) Obtain the general PMF for the random variable X + Y . Hint: Consider the set A = {(x, y) : x + y = m} = {(m, 0), (m1, 1), . . . , (0, m)}. Sum the joint PMF over (x, y) A and use the binomial theorem, which says that m \u0012 \u0013 X m k mk = (a + b)m a b k k=0 2