Problem 21E

The lifetime of a certain component, in years, has probability density function

Two such components, whose lifetimes are independent, are available. As soon as the first component fails, it is replaced with the second component. Let X denote the lifetime of the first component, and let Y denote the lifetime of the second component.

a. Find the joint probability density function of X and Y.

b. Find P(X ≤ 1 and Y > 1).

c. Find μx.

d. Find μx+Y.

e. Find P(X + Y ≤ 2). (Hint: Sketch the region of the plane where x + y ≤ 2, and then integrate the joint probability density function over that region.)

Answer :

Step 1 of 6:

Given,

The probability density function of the lifetime of a certain component, in years.

We have two components, whose lifetimes are independent. As soon as the first component fails, it replaces with second component.

Let, X = lifetime of the first component. And Y = lifetime of the second component.