Two different professors have just submitted final exams for duplication. Let X denote the number of typographical errors on the first professor's exam and Y denote the number of such errors on the second exam. Suppose X has a Poisson distribution with parameter p, Y has a Poisson distribution with parameter , and X and Y are independent.

a. What is the joint pmf of X and Y?

b. What is the probability that at most one error is made on both exams combined?

c. Obtain a general expression for the probability that the total number of errors in the two exams is m (where m is a nonnegative integer). [Hint: A = {(x, y): x+y=m} ((m, 0), (m1, 1), (1, m-1), (0, m)}. Now sum the joint pmf over (x, y) E A and use the binomial theorem, which says that for any

a, b.] m 1-0 m (ab abm-k = = (a + by