Solution Found!
Let X and Y be Bernoulli random variables. Let Z = XY.a.
Chapter 4, Problem 7E(choose chapter or problem)
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
Questions & Answers
QUESTION:
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: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.