# 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}.

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 ${P}_{Z}={P}_{X}{P}_{Y}$.
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Jaylen Fountain
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

Let us consider $P\left(Z=1\right)$

As both are independent

${P}_{Z}={P}_{X}\ast {P}_{Y}$
Hence proved.