11. Two different professors have just submitted nal exams for duplication. Let X denote the number of typographical errors on the rst professor s exam and Y denote the number of such errors on the second exam. Suppose X has a Poisson distribution with parameter l, Y has a Poisson distribution with parameter u, 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) H A and use the binomial theorem, which says that for any

a, b.]