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 , 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), (m  1, 1), . . . , (1, m  1), (0, m)}. Now sum the joint pmf over (x, y)  A and use the binomial theorem, which says that

m k
0

 ak bmk
(a b)m for any

a, b.]