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Let X and Y be Bernoulli random variables. Let Z = XY.a.

Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi ISBN: 9780073401331 38

Solution for problem 7E Chapter 4.1

Statistics for Engineers and Scientists | 4th Edition

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Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi

Statistics for Engineers and Scientists | 4th Edition

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Problem 7E

Let \(X\) and \(Y\) be Bernoulli random variables. Let Z = XY.

a. Show that \(Z\) is a Bernoulli random variable.

b. Show that if \(X\) and \(Y\) are independent, then \(p_{Z}=p_{X}p_{Y}\)

Equation Transcription:

Text Transcription:

X

Y

Z

p_Z = p_X p_Y

Step-by-Step Solution:

Answer:

Step 1 of 2:

(a)

In this question, we are asked to prove whether is a Bernoulli random variable or not.

Let  and  be Bernoulli random variables. Let  = .

Case 1:

If and ,

Case 2:

If and ,

Case 2:

If and ,

The random variable  is said to have the Bernoulli distribution when random event results in the success, then . Otherwise .

So from the definition we can say that  is also a Bernoulli random variable.

Since possible values of the product  = are 0 and 1 from the cases we have seen.

Hence proved.


Step 2 of 2

Chapter 4.1, Problem 7E is Solved
Textbook: Statistics for Engineers and Scientists
Edition: 4
Author: William Navidi
ISBN: 9780073401331

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Let X and Y be Bernoulli random variables. Let Z = XY.a.