Question

A system contains two components X, Y which both need to work in order for the system to run. The lifetime of component X is an exponential random variable X with parameter 2, and the lifetime of component Y is an exponential random variable Y with parameter 1. Assume that X,Y are independent. Let Z denote the lifetime of the system, which depends on X, Y

a. Describe Z as a function of X, Y

b. Find the PDF of Z

c.Find E[Z]

Answer 1

a) We are given here that both X and Y need to work for the system to work. Therefore the lifetime Z here would be given as the lifetime of min component. Therefore Z here is defined as:

Z = min(X, Y)

b) The CDF for Z is first obtained here as:

F(z) = P(Z <= z) = 1 - P(Z > z) = 1 - P(X > z)P(Y > z)

Using the CDF for X and Y now, we have here:

-3 z P(Z <=)=1-e

Therefore Z is an exponential distribution with parameter 3 and so the PDF for Z here is given as:

f(s) = Зе- -32,0< Z<ю

c) The expected value of an exponential distribution is reciprocal of its parameter.

Therefore E(Z) = 1/3 here.