Question

A device contains two circuits. The second circuit is a backup for the first, so the second is used only when the first has failed. The device fails when and only when the second circuit fails. Let X and Y be the times (in years) at which the first and second circuits fail, respectively. Their joint probability density function (pdf) is 6e-xe-20 for 0 < x < y 〈 oo; otherwise (a) Find the pdf of X. Identify the distribution of X (b) Find the pdf of Y (c) What is the expected time at which the device fails?

Answer 1

Explanation:

The pdf of X is given by 3e^x(1-e^-2y). Please check the screenshot for the workings.

The pdf of Y is given by 6e^-2y. Please check the screenshot for the workings.

The expected time the device fails is the expected time Y fails, so we are going to find the expected time of Y which equals to 0.83.

Please check the screenshop for the workings. Thanks