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# A production facility contains two machines that are used ISBN: 9780073401331 38

## Solution for problem 18E Chapter 2.6

Statistics for Engineers and Scientists | 4th Edition

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Problem 18E

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.

Step-by-Step Solution:

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 0.75 dx + 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

Step 3 of 3

##### ISBN: 9780073401331

This full solution covers the following key subjects: Probability, machines, Density, use, machine. This expansive textbook survival guide covers 153 chapters, and 2440 solutions. The full step-by-step solution to problem: 18E from chapter: 2.6 was answered by , our top Statistics solution expert on 06/28/17, 11:15AM. This textbook survival guide was created for the textbook: Statistics for Engineers and Scientists , edition: 4. Since the solution to 18E from 2.6 chapter was answered, more than 1146 students have viewed the full step-by-step answer. Statistics for Engineers and Scientists was written by and is associated to the ISBN: 9780073401331. The answer to “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.” is broken down into a number of easy to follow steps, and 94 words.

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