Let $X$ be a normally distributed random variable, with mean $\mu$ and variance $\sigma^2$. Consider a random vector $$V = \left[ X^n, X^{n-1}, \dots, X^2, X, 1 \right]^T $$ What is the expected value of $V$, $\bar{V}$, and covariance matrix $\Sigma$? Do they have any nice properties?
Wikipedia has a listing of the the first eight non-central moments of a normal distribution, which I believe should be the expected values. Is that right?