Discovering Statistics

A den for Learning

2D-Random Variables


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.

(r,s)-th order Raw Moments

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

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

(r,s)-th order Central Moments

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)
\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) &

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