2 Problems on Applied Probability and Statistics I - Assignment 8 | STAT 400, Assignments of Probability and Statistics

Download 2 Problems on Applied Probability and Statistics I - Assignment 8 | STAT 400 and more Assignments Probability and Statistics in PDF only on Docsity!

Stat 400 In Class Worksheet 8 T.A. Emily King April 2, 2008

1. (Modified from book p.194 # 3) A certain market has both an express checkout line and a superexpress checkout line. Let X 1 denote the number of customers in line at the express checkout at a particular time of day, and let X 2 denote the number of customers in line at the superexpress checkout at the same time. Suppose the joint pmf of X 1 and X 2 is given in the accompanying table: (x 1 , x 2 ) 0 1 2 3 0. 08. 07. 04. 00 1. 06. 15. 05. 04 2. 05. 04. 10. 06 3. 00. 03. 04. 07 4. 00. 01. 05. 06 (a) What is P (X 1 = 1, X 2 = 1), that is, the probability that there is exactly one customer in each line? (b) What is P (X 1 = X 2 ), that is, the probability that the numbers of custmoers in the two lines are identical? (c) Let A denote the event that there are at least two more customers in one line than in the other line. Express A in terms of X 1 and X 2 , and calculate the probability of this event. (d) What is the probability that the total number of customers in the two lines is exactly four? At least four? (e) What are the marginal pmf’s of X 1 and X 2? (f) Are X 1 and X 2 independent rv’s? Explain.

2. (Book p. 181 # 111): The mode of a continuous distribution is the value X∗^ that maximizes f (x). (a) What is the mode of a normal distribution with parameters μ and σ. (b) Does the uniform distribution with parameters A and B have a single mode? Why or why not? (c) What is the mode of an exponential distribution with parameter λ?