2D-Random Variables

BIVARIATE MOMENTS

Let (X,Y) be a bivariate random variable and g(x,y) be a real valued function of the two variables then,

If (X,Y) is of discrete type with pmf $Pr[X=x_i,Y=y_j] = p_{ij}$ then,

E \left(g(X,Y) \right) = \sum_{i=1}^\infty \sum_{j=1}^{\infty} g(x_i,y_j)p_{ij}

provided the series converges.

If (X,Y) is of continuous type with pdf f(x,y) then,

E \left(g(X,Y) \right) = \int_{-\infty}^\infty \int_{\infty}^{\infty} g(x,y)f(x,y)dxdi

provided that the integral converges absolutely.

The (r/s)-th order raw moment of (X,Y), if it exists is given by:

\mu'_{r,s} = E(X^rY^s)

\begin{matrix} \mu'_{1,0}=E(X) & \mu'_{2,0}=E(X^2) \\ \mu'_{0,1}=E(Y) & \mu'_{0,2}=E(Y^2) \\ \mu'_{1,1} = E(XY) & \end{matrix}

The (r/s)-th order central moment of (X,Y), if it exists is given by:

\mu_{r,s} = E\left((X-E(X))^r(Y-E(Y))^s\right)

\begin{matrix} \mu_{0,0} =1 \\ \mu_{1,0}=\mu_{0,1}=0 \\ \mu_{2,0}=V(X) \\ \mu_{0,2}=V(Y) \\ \mu_{1,1} = cov(X,Y) & \end{matrix}