Let $X, Y$ and $Z$ be three independent random variables such that $E(X)=E(Y)=E(Z)=0$ and $Var(X)=Var(Y)=Var(Z)=1$. Calculate $E[(X^2)(Y+5Z)^2]$
I know that the answer is $26$.
Since all of the expected values of $X, Y$ and $Z$ are all the same, I have replaced each expected value of $X, Y$ and $Z$ with just $E[X]$. For example, $E[(X^2)(Y^2)]$ is now $(E[X])^4$.
Doing this, I'm left with $26E[X^4]$.
Since the variance of each random variable is one, I know I need to somehow turn $E[X^4]$ into the formula for variance…. Thanks guys
Best Answer
The random variables $X,Y,Z$ are mutually independent, and have identical expectation, of $0$, and variance, of $1$. This does not mean they are interchangable; they are still three distinct variables.
For instance, $X$ is independent from $Y$, but clearly not independent from $X$.
Because $\mathsf {Cov}(X,Y)=\mathsf E(XY)-\mathsf E(X)\mathsf E(Y)$ , $\mathsf {Cov}(X,Y)=0$, $\mathsf E(X)=0$, and $\mathsf E(Y)=0$, therefore $\mathsf E(XY)=0$.
Because $\mathsf {Var}(X)=\mathsf E(X^2)-\mathsf E(X)^2$ , $\mathsf {Var}(X)=1$, and $\mathsf E(X)=0$, therefore $\mathsf E(X^2)=1$ .
So obviously $\mathsf E(X^2)\neq \mathsf E(XY)$ and so forth.
So we have : $\mathsf E[(X^2)(Y+5Z)^2]~{=\mathsf E(X^2)~\mathsf E(Y^2+10YZ+25Z^2)\\~\vdots\\ = 26}$