I want to obtain equations for expected value and variance for a homogeneous polynomial of arbitrary order $n$.
For a quadratic form, which is a special case of a homogeneous polynomial, there are known equations for expected value and variance:
$ E[A^T HA] = tr[H\Sigma]+\mu^TH\mu, $
$ var[A^T HA] = 2tr[H\Sigma H\Sigma]+4\mu^TH\Sigma H\mu, $
where $A$ is a vector of $M$ random variables, $H$ is a square symmetric matrix with polynomial coefficients and $\mu$ and $\Sigma$ are the expected value and variance matrix of $A$, respectively.
I want to consider a homogeneous polynomial of order $n$:
$ \sum_{i_1=1}^{M} \sum_{i_2=1}^{M} ... \sum_{i_n=1}^{M} h^{(n)}(i_1, i_2, ...,i_n)a_{i_1}a_{i_2} ... a_{i_n},$
where $a_i$ is a random variable from vector $A$ and $h^{(n)}$ are polynomial coefficients. E.g. a 3-rd order polynomial for $M=3$ will look like this:
$ h_1^{(3)} a_1^3 + h_2^{(3)} a_1^2 a_2 + h_3^{(3)} a_1^2 a_3+ h_4^{(3)} a_1 a_2^2 + h_5^{(3)} a_1 a_2 a_3+ h_6^{(3)} a_1 a_3^2 + h_7^{(3)} a_2^3 + h_8^{(3)} a_2^2 a_3 + h_9^{(3)} a_2 a_3^2 + h_{10}^{(3)} a_3^3 . $
Question: How to derive equations for expected value and variance of such polynomial or where can I look them up, if they are widely known/used?
I don't understand what the polynomials are, but there is a general formula for the mean and variance of a polynomial function of a random variable.
Let $$p(x)=\sum_{i=0}^na_i x^i$$ be a given polynomial.
The square of the polynomial is $$[p(x)]^2=\sum_{i=0}^{2n} b_i x^i$$ where $$b_i=\sum_{j=0}^{i}a_j a_{i-j}$$
Also, let $X$ be a random variable with moments about the origin $$\mu'_k=E[X^k]$$ for $k=0, 1, 2, ...$
Finally, let $Y=p(X)$.
Then, the mean and variance of $Y$ are: $$E[Y]=\sum_{i=0}^na_i \mu'_i$$ and $$Var[Y]=E[Y^2]-(E[Y])^2=\sum_{i=0}^{2n} b_i \mu'_i-\left(\sum_{i=0}^na_i \mu'_i \right)^2$$