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}.

Tolnaio 2021-03-02 Answered
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 PZ=PXPY.
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Expert Answer

Jaylen Fountain
Answered 2021-03-03 Author has 170 answers
Step 1
Given
X and Y are Bernoulli random variables.
Bernoulli random variables are discrete random variables which have outcomes 0 or 1.
Z=XY
As the possible values of X and Y are 0 or 1. the possible values of Z=XY , for all values of X and Y are also 0 or 1.
Hence Z=XY is also a Bernoulli random variable.
Step 2
b)Given X and Y are independent events
P(X and Y)=P(X)P(Y)
Let us consider P(Z=1)
P(Z=1)=P(X=1 and Y=1)
As both are independent
P(Z=1)=P(X=1 and Y=1)=P(X=1)P(Y=1)
PZ=PXPY
Hence proved.
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