# Moment Generating Function, Marginal & Conditional Distribution

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.

#### Solution Summary

Moment Generating Function, Marginal and Conditional Distributions and Probability Density Functions are investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.