the sum of two independent exponentials is a gamma
You want P(X+Y>3).
I want to confirm that after finding the convolution of the independent variables I need to use the gamma distribution to solve the following problem:
A Mars probe has two batteries. Once a battery is activated, its future lifetime is exponential with mean 1 year. The first battery is activated when the probe lands on Mars. The second battery is activated when the first fails. Battery lifetimes after activation are independent. The probe transmits data until both batteries have failed. Calculate the probability that the probe is transmitting data three years after landing.
Using MGFs we can easily prove that
(where they are independent), then
This is very useful for adding Chi-Squares, which is important in Design of Experiments, where
we use Cochran's Theorem repeatedly.