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, and let Y be the number of hours that the second machine is in use, on a randomly chosen day. Assume that X and Y have joint probability density function given by

a. What is the probability that both machines are in operation for more than half an hour?

b. Find the marginal probability density functions fX(x) and fY(y).

c. Are X and Y independent? Explain.

Answer :

Step 1 of 4 :

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

Step 2 of 4 :

The claim is to find the probability that both machines are in operation for more than half an hour.That is , P( X > 0.5 and Y> 0.5 ) = (+ ) dy dx

First we have to integrate in terms of y

(+ ) dy

Take constant out, (+ ) dy

Then apply sum rule,

f(x) g(x) dx = f(x) dx g(x) dx

Thus, dy+ dy

dy = y and dy = ( from the power rule dx = )

Therefore, (+ ) dy = ( y + ) + c

Then compute the limits

For limit 0.5 ( y + ) = 0.75+ 0.0625001

For limit 1 ( y + ) = ( 3+ 1 )

Therefore, (+ ) dy = ( 3+ 1 ) - (0.75+ 0.0625001)

= 0.75+ 0.4375

Integrate with respect to x

(0.75+ 0.4375) dx

By applying the sum rule 0.75dx + 0.4375 dx

0.75= and 0.4375 dx = 0.4375 x

Therefore, (0.75+ 0.4375) dx = ( + 0.4375 x) + C

For limit 0.5 + 0.4375 x = 0.25

For limit 1 + 0.4375 x = 0.6875

(0.75+ 0.4375) dx = 0.6875 - 0.25

= 0.4375

Therefore, the probability that both machines are in operation for more than half an hour is 0.4375