Given a random variable $X$, a location scale transformation of $X$ is a new random variable $Y=aX+b$ where $a$ and $b$ are constants with $a>0$.
The location scale transformation $aX+b$ horizontally scales the distribution of $X$ by the factor $a$, and then shifts the distribution so obtained by the factor $b$ on the real line $\mathbb{R}$.
- In an intuitive sense, the expected value $\mathbb{E}[X]$ of a random variable is the center of mass of the distribution of $X$. Shifting the distribution of $X$ by a factor $b$, shifts the center of mass by the factor $b$. Scaling the distribution of $X$ by a factor $a$, scales the center of mass by $a$. In other words, $$\mathbb{E}[aX+b]=a\mathbb{E}[X]+b$$
- Similarly, the variance of $X$ is a measure of the horizontal spread of the distribution of $X$, but the $\text{Var}[X]$ is defined as squared-distance. Thus scaling the distribution of $X$ by a factor $a$, scales the $\text{Var}[X]$ by the factor $a^2$. Shifting the distribution of $X$ by any factor will not affect the spread of distribution, $\text{Var}[X]$, but only affects center of mass. In other words, $$\text{Var}[aX+b]=\text{Var}[aX]=a^2\text{Var}[X]$$
Now, here is an hint to your problem: $Y=\dfrac{X-\mu}{\sigma}=\dfrac{1}{\sigma}X-\dfrac{\mu}{\sigma}$, which can be written as $aX+b$. Find $a$ and $b$, and then use the location-scale transformation.
I cannot give you one expression, but here are several articles that cover some non-normal cases:
Browne, M. W., & Shapiro, A. (1986). The asymptotic covariance matrix of sample correlation coefficients under general conditions. Linear Algebra and its Applications, 82, 169-176.
Gayen, A. K. (1951). The frequency distribution of the product-moment correlation coefficient in random samples of any size drawn from non-normal universes. Biometrika, 38, 219-247.
Kowalski, C. (1972). On the effects of non-normality on the distribution of the sample product-moment correlation coefficient. Applied Statistics, 21, 1-12.
Subrahmaniam, K., & Gajjar, A. V. (1980). Robustness to nonnormality of some transformations of the sample correlation coefficient. Journal of Multivariate Analysis, 10, 60-77.
Yuan, K.-H., & Bentler, P. M. (2000). Inferences on correlation coefficients in some classes of nonnormal distributions. Journal of Multivariate Analysis, 72, 230-248.
Best Answer
Since $X^TAX$ is a scalar, $$\text{Tr}(X^TAX)= X^TAX = \text{Tr}(AXX^T)$$ so that $$\text{E}(X^TAX) = \text{E}(\text{Tr}(AXX^T)) = \text{Tr}(\text{E} (A XX^T)) = \text{Tr}(A\text{E}(XX^T))$$.
Here we have used that the trace of a product are invariant under cyclical permutations of the factors, and that the trace is a linear operator, so commutes with expectation. The variance is a much more involved computation, which also need some higher moments of $X$. That calculation can be found in Seber: "Linear Regression Analysis" (Wiley)