Question

NASA is considering a switching (cold standby) system of components with the following distribution

f(x) = xe^-x

(x in days). However the switch is not perfect, and will fail with probability .2 when called upon. Once it fails it will not work again. NASA wants to know what is the mean lifetime and the variance of the lifetime for a 5 component system. The would also like to know the probability that such a system will last 3 days.

Answer 1

Here, X = no. of components failing and p = probability of a component failing. Also, n = 5 components.
Hence, X ~ Binomial(n = 5, p = 0.20).
Hence, mean lifetime = E(X) = n * p = 5 * 0.20 = 1.
Variance of the lifetime = Var(X) = n * p * (1-p) = 5 * 0.2 * 0.8 = 0.8.

Prob.(such a system will last for 3 days) = f(3) = 3 * exp(-3) = 0.1494.