Solution Found!
Let X and Y be Bernoulli random variables. Let Z = X+ Y.a.
Chapter 4, Problem 4E(choose chapter or problem)
Let \(X\) and \(Y\) be Bernoulli random variables. Let Z = X + Y.
a. Show that if \(X\) and \(Y\) cannot both be equal to 1, then \(Z\) is a Bernoulli random variable.
b. Show that if \(X\) and \(Y\) cannot both be equal to 1, then \(p_{Z}=p_{X}+p_{Y}\).
c. Show that if \(X\) and \(Y\) can both be equal to 1, then \(Z\) is not a Bernoulli random variable.
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 = X + Y.
a. Show that if \(X\) and \(Y\) cannot both be equal to 1, then \(Z\) is a Bernoulli random variable.
b. Show that if \(X\) and \(Y\) cannot both be equal to 1, then \(p_{Z}=p_{X}+p_{Y}\).
c. Show that if \(X\) and \(Y\) can both be equal to 1, then \(Z\) is not a Bernoulli random variable.
Equation Transcription:
Text Transcription:
X
Y
Z
p_Z = p_X + p_Y
ANSWER:Answer:
Step 1 of 3:
(a)
In this question, we are asked to prove that if and
cannot both be equal to 1, then
is a Bernoulli random variable.
Let and
be Bernoulli random variables. Let
=
.
Case 1:
If and
,
Case 2:
If and
,
So in both the cases .
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 value of .
Hence proved.