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

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.