Two components of a minicomputer have the following 5.1 Jointly Distributed Random Variables 211 a. b. c. What is the joint pmf of X and Y? What is the probability that at most one error is made on both exams combined? 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 5 {(x, y): x 1 y 5 m} 5 {(m, 0), (m 2 1, 1), . . . , (1, m 2 1), (0, m)}. Now sum the joint pmf over (x, y) [ A and use the binomial theorem, which says that om 1m2akbm2k 5 (a 1 b)m k50 k joint pdf for their useful lifetimes X and Y: 5xe2x(11y) x $ 0 and y $ 0 a. b. c. What is the probability that the lifetime X of the first component exceeds 3? What are the marginal pdf’s of X and Y? Are the two lifetimes independent? Explain. What is the probability that the lifetime of at least one component exceeds 3?