Product of Moments

The rth and sth product of Moments about the means on the random variables $X$ and $Y$ ( $μ_{r, s}$ ) is

\begin{aligned} μ_{r, s} & = E [(X - μ_{X})^{r} (Y - μ_{Y})] \\ = \sum_{x} \sum_{y} (x - μ_{X})^{r} (y - μ_{Y})^{s} \cdot f_{X, Y} (x, y) \\ or for continuous: \\ μ_{r, s} & = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} (x - μ_{x})^{r} (y - μ_{y})^{s} \cdot f_{X, Y} (x, y) d x d y \\ (r, s) \in N^{2} \end{aligned}

When $r = s = 1, μ_{1, 1} = E [(X - μ_{X}) (Y - μ_{Y})]$ is the Covariance of $X$ and $Y$ , $c o v (X, Y)$ or $σ_{X Y}$

The rth and sth product of moment (about the origin) of the random variables $X$ and $Y$ ( $μ_{r, s}^{'}$ ) is

\begin{aligned} μ_{r, s}^{'} & = \sum_{x} \sum_{y} x^{r} y^{s} \cdot f_{X, Y} (x, y) = E [X^{r} Y^{s}] \\ or for continuous: \\ μ_{r, s}^{'} & = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} x^{r} y^{s} \cdot f_{X, Y} (x, y) d x d y \\ (r, s) \in N^{2} \end{aligned}

If $X$ and $Y$ are independent: