Refer to Exercise 18.

a. Find Cov(X, Y).

b. Find ρX,Y.

c. Find the conditional probability density function fY|X(y|0.5).

d. Find the conditional expectation E(Y | X =0.5).

Exercise 18: A production facility contains two machines that are used to rework items that are initially defective. Let X be the number of hours that the first machine is in use, let Y be the number of hours that the second machine is in use, on a randomly chosen day. Assume tyhat X and Y have joint probability density function given by

What is the probability that both machines are in operations for more than half an hour?Find the marginal probability density functions and Are X and Y independent? Explain.Answer :

Step 1 of 5 :

Given, A production facility contains two machines that are used to rework items that are initially defective.

Let, X = number of hours that the first machine is in use and

Y = number of hours the second machine is in use

Let X and Y have the joint probability density function