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 3, and the lifetime of component Y is an exponential random variable Y with parameter 2. 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) As we are given that both components need to work for the system to work, therefore the minimum of the X and Y values would be the lifetime of Z.

Therefore, Z = min(X, Y) here.

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

Using the PDF for exponential distributions, we have here:

This is a CDF for an exponential distribution with parameter 5

Therefore the PDF for Z here is given as:

this is the required PDF for Z here.

c) The expected value of Z is computed as the reciprocal of its parameter computed as:

E(Z) = 1/5 = 0.2

Therefore 0.2 is the required expected value here.