Tolnaio

2021-03-02

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

a) Show that Z is a Bernoulli random variable.

b) Show that if X and Y are independent, then

Jaylen Fountain

Skilled2021-03-03Added 169 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

$P(X\text{}and\text{}Y)=P(X)\ast P(Y)$

Let us consider$P(Z=1)$

$P(Z=1)=P(X=1\text{}and\text{}Y=1)$

As both are independent

$P(Z=1)=P(X=1\text{}and\text{}Y=1)=P(X=1)\ast P(Y=1)$

${P}_{Z}={P}_{X}\ast {P}_{Y}$

Hence proved.

Given

X and Y are Bernoulli random variables.

Bernoulli random variables are discrete random variables which have outcomes 0 or 1.

As the possible values of X and Y are 0 or 1. the possible values of

Hence

Step 2

b)Given X and Y are independent events

Let us consider

As both are independent

Hence proved.

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