Chapter 5.1, Problem 11E

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 µ₁, Y has a Poisson distribution with parameter µ₂, 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) $\in$ A and use the binomial theorem, which says that

     ${\displaystyle \underset{k=0}{\overset{m}{\sum }}\left(\begin{array}{c}m\\ k\end{array}\right)}{a}^k{b}^{m-k}={(a+b)}^m$

for any a, b.]