Answer Happy

Let 𝑋 and 𝑌 be the lifetimes (measured in years) of Machine A
and Machine B, respectively. Assume that 𝑋 is continuous and
uniformly distributed in [1, 4] and 𝑌 is exponentially distributed
with the mean being 2 years. Further assume that the lifetimes of
Machine A and Machine B are independent.
(a) Find the probability that the lifetime of Machine A is at
least 1 year longer than the lifetime of Machine B.
(b) Find the probability that the lifetimes of both Machine A
and Machine B are less than 2 years.
(c) Find the probability that the sum of the lifetimes of
Machine A and Machine B is greater than 4 years.
3. (9 pts) Let X and Y be the lifetimes (measured in years) of Machine A and Machine B, respectively. Assume that X is continuous and uniformly distributed in [1, 4] and Y is exponentially distributed with the mean being 2 years. Further assume that the lifetimes of Machine A and Machine B are independent. (a) Find the probability that the lifetime of Machine A is at least 1 year longer than the lifetime of Machine B. (b) Find the probability that the lifetimes of both Machine A and Machine B are less than 2 years. (c) Find the probability that the sum of the lifetimes of Machine A and Machine B is greater than 4 years.