Variance and Conditional Expectation of combination of independent random variables

Question

Let $X$, $Y$ be two independent standard normal random variables, $X \sim N(0, 1)$ and $Y\sim N(0, 1)$. Let $Z$ be a random variable defined by $Z = XY+2X$.

I have to find $\text{Var}(Z)$ and $\text{E}(Z|X)$.

$\text{Var}(Z)=\text{Var}(XY)+4\text{Var}(x)+2\text{Cov}(XY,Y)$

$\text{E}(Z|X)=\text{E}(XY|X)+2\text{E}(X|X)$

but still I don't know how to proceed further.

Can someone answer please? (Probably my question is very silly)

score 0 · Accepted Answer · answered Nov 08 '17 at 17:42

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Hints: $$ \operatorname{Var}Z=\operatorname EZ^2-(\operatorname EZ)^2 $$ $$ \operatorname E[XY]=\operatorname EX\operatorname EY $$ $$ \operatorname E[X^2Y^2]=\operatorname EX^2\operatorname EY^2 $$ $$ \operatorname E(X\mid X)=X $$ $$ \operatorname E(XY\mid X)=X\operatorname E(Y\mid X) $$ $$ \operatorname E(Y\mid X)=\operatorname EY $$

answered Nov 08 '17 at 17:42

Cm7F7Bb

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So if X and Y are independent, can I say that independence is invariant under non linear transformation? i.e. E[X$^2$Y]=E[X$^2$]E[Y] – Francesco Lorenzo Flamini Nov 08 '17 at 18:42
@FrancescoLorenzoFlamini Yes (see here). – Cm7F7Bb Nov 10 '17 at 15:38

Variance and Conditional Expectation of combination of independent random variables

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