1. If the moment generating function ( mgf ) of X is
(a) Find the mean of X.
(b) Find the variance of X.
(c) Find the pdf of X.
2. The joint probability mass function for random variables X and Y is:
FXY (x, y) =(x + y)/32; x = 1, 2; y = 1, 2, 3, 4
(a) Show that fXY is a valid mass function.
(b) Find the marginal distributions of X and Y.
(c) Find the conditional distributions of X|Y and Y |X.
(d) Find P(X > Y), P(Y = 2X), P(X + Y = 3), P(X 3 − y).
(e) Are X and Y dependent or independent? Explain why.
(f) Find means μX and μY.
(g) Find variances,
.
(h) Find covariance and correlation coefficient,
3. The joint probability density function for random variables X and Y is:
(a) Sketch fXY.
(b) Find the marginal distributions of X and Y.
(c) Find the conditional distributions of X|Y and Y |X.
(d) Find
(e) Are X and Y dependent or independent? Explain why.
(f) Find means, μX and μY.
(g) Find variances
(h) Find covariance and correlation coefficient,
4. The times that a cashier spends processing each person's transaction are independent and identically distributed random variables with a mean of 1.5 minutes and standard deviation of 1 minute.
(a) What is the approximate probability that the orders of 100 people can be processed in less than 2 hours?
(b) Find the number of customers, n, such that the probability that the orders of all n customers can be processed in less than 2 hours is approximately 0.9.