Consider $Z_1$and $Z_2$, two independent random variables that both follow the normal distribution $N(\mu, \sigma ^2)$:
$X = Z_2$
$Y = Z_1 + Z_1(Z_2 - \mu)^p$ , where $p$ is a positive integer.
Recall that if $X ∼ N(\mu, \sigma ^2)$, then:
- $\mu_{2k} = \mathbb{E}[(X-\mu)^{2k}] = \frac{(2k)!}{2^kk!}\sigma^{2k} $
- $\mu_{2k+1} = \mathbb{E}[(X-\mu)^{2k+1}] = 0 $
I have to find the best linear predictor of $Y|X=x$ and therefore, I need to find $\operatorname{Cov}(X,Y)$. After some manipulations when calculating $\mathbb{E}[XY]$, I end up dealing with this term $\mathbb{E}[Z_2(Z_2-\mu)^p)]$:
$\mathbb{E}[XY] = \mathbb{E}[Z_2(Z_1 + Z_1(Z_2 - \mu)^p)]$
$ = \mathbb{E}[Z_1Z_2] + \mathbb{E}[Z_1Z_2(Z_2 - \mu)^p]$ ; $Z_1$ and $Z_2$ are independent so $\mathbb{E}[Z_1Z_2] = \mathbb{E}[Z_1]\mathbb{E}[Z_2] = \mu^2$
$= \mu^2 + \mathbb{E}[Z_1]\mathbb{E}[Z_2(Z_2 - \mu)^p]$
$= \mu^2 + \mu \mathbb{E}[Z_2(Z_2 - \mu)^p]$
and I am not sure how to manipulate this to get what I want, am I missing something?
$$E[Z_2(Z_2-\mu)^p]=E[(Z_2-\mu+\mu)(Z_2-\mu)^p]=E[(Z_2-\mu)^{p+1}]+\mu E[(Z_2-\mu)^{p}]$$