Find the best predictor and the best linear predictor of $Y^2$ given $X$. Suppose $(X, Y ) \sim N(0, 0, 1, 1, p ).$

Once more, there's another question that I'm clueless on how to start. I should have dropped this course earlier.

Suppose $(X, Y ) \sim BN(0, 0, 1, 1, p )$, meaning that $X$ and $Y$ are bivariate normal with zero means and unit variances and correlation $p$ . Find the best predictor and the best linear predictor of $Y^2$ given $X$ and and their respective mean squared prediction errors.

Basically I'm supposed to find $E(Y^2 |X)$ as the best predictor given $X$

$Y^2= \alpha+\beta X$ as the best liner predictor where $\beta = \text{Cov}(Y^2 ,X) / \text{Var} (X)$ and then to find the mean squared prediction errors.

$E(Y^2 |X)$ was easy to find and so was $\text{Var} (X)$

But when doing $\text{Cov}(Y^2 ,X)$ I got $P^2 E(X^3)$. And I don't know what is $E(X^3)$ To find Mean squared prediction errors I used $E(Y^2 -E(Y^2 |X))2$ which i broke it down into $E(Y^4 )-E(E(Y|X)2 )$.

From there I'm stuck. I really have no clue on how to proceed. I could integrate using the density of a bivariate normal distribution but the integration looks messy and I keep getting stuck.

• If X is standard normal then E(X^3)=0. If Y is standard normal then E(Y^4)=3. Are these answering your problem? – Did Mar 22 '14 at 8:52
• A bit, but how did we get those numbers though is what I'm more interested in. – Thatguy Mar 22 '14 at 9:04
• (Why did you delete your previous comment?) Actually you should know them, otherwise: en.wikipedia.org/wiki/Normal_distribution#Moments – Did Mar 22 '14 at 9:44
• Ah thanks, We only just covered moments but only of Expectation and Variance. Thanks for the link. And did I delete a comment? I don't think I did so purposely. – Thatguy Mar 22 '14 at 9:59
• Yeah, it started by something similar to "is it so easy...". – Did Mar 22 '14 at 10:05