Finding Var(XYZ)

Question

Let $X$, $Y$, $Z$ be independent and identically distributed random variables with mean = 1 and variance = 2. Find $Var(XYZ)$.

$Var(XYZ)$

$= E((XYZ)^2) - (E(XYZ))^2$

$= E(X^2Y^2Z^2) - (E(XYZ))^2$

$= E(X^2)E(Y^2)E(Z^2) -(E(X)E(Y)E(Z))^2$ since $X$, $Y$, $Z$ are independent.

$= E(X^2)E(Y^2)E(Z^2) -(1\cdot1\cdot1)^2$

$= E(X^2)E(Y^2)E(Z^2) - 1$

$=$ ..... ?

$= 26$

Answer 1

Note that $$ 2=\text{Var}(X)=EX^2-(EX)^2=EX^2-1 $$ Thus $$ EX^2=EY^2=EZ^2=3 $$ since $X,Y,Z$ are identical in distribution.