■ Joint Moment About the Origin

[eqn: &m_nk ``=`` E[`X^n ` Y^k `]``=`` INT _{ - INF }^{ INF }`` INT _{ - INF }^{ INF }`` x^n ` y^k ` f_X,Y ` (x,y) dx`dy #& m_10 ``=`` E[`X`]``=``bar X #& m_01 ``=`` E[`Y`]``=``bar Y]

[eqn: E[X], ```E[Y]] are the expected values of X and Y

.

■ Correlation of X and Y

[eqn: R_XY ``=`` m_11 ``=`` E[`X`Y`]``=`` INT _ {- INF }^{INF } INT _ {- INF }^{INF } x`y`f_X,Y (x,y) `dx`dy]

■ Uncorrelated

[eqn: R_XY ``=`` E[`X`]`E[`Y`]]

■ Statistical Independece of Random Variables X, Y

[eqn: P`LEFT{`X <= x , `Y <= y RIGHT}``=``P LEFT{`X<=x RIGHT}`P LEFT{`Y<=y RIGHT}]

[eqn: F_X,Y (x,y)``=``F_X (x)`F_Y (y)]

[eqn: f_X,Y (x,y)``=``f_X (x) `f_Y (y)]

Statistical independece of X and Y is sufficient to guarantee they are uncorrelated. The converse of this statement, that is, that X and Y are independent if X and Y are uncorrelated, is not necessarity true in general. Uncorrelated gaussian random variables are, however, known to also be independent.

■ Joint Central Moment

[eqn: mu -_nk``=`` &E[`(X`-`bar X )^n`(Y`-`bar Y )^k ] #& INT _{ - INF }^{INF } INT _{ - INF }^{INF } (x - bar X )^n` (y - bar Y )^k f_X,Y (x,y)`dx`dy]

[eqn: & mu _20 ``=``E[(X-bar X )^2 ]``=`` sigma _X ^2 #& mu _02 ``=``E[(Y-bar Y )^2 ]``=`` sigma _Y ^2]

The second-order contral moments [eqn: mu_20 ,``` mu_02] are the variances of X and Y.

■ Covariance of X and Y

[eqn: C_XY``=``mu_11``&=``E[(X-bar X )(Y-bar Y )] # &=`` INT _{ - INF }^{INF } INT _{ - INF }^{INF } (x-bar X )(y-bar Y )f_X,Y (x,y)`dx`dy]

[eqn: C_XY ``=``R_XY ``-`` barX` barY``=``R_XY``-``E[X]`E[Y]]

[eqn: C_XY ``=``0]   X and Y independent or uncorrelated

[eqn: C_XY``=``-`E[X]`E[Y]]      X and Y orthogonal

■ Correlation Coefficient of [eqn: X] and [eqn: Y]

[eqn: rho ``=`` mu_11 `/` SQRT {mu_20 mu_02 } ``=``C_XY``/``sigma_X` sigma_Y]

[eqn: rho``=``E[{(X-bar X )} over {sigma_X } {(Y-bar Y )} over {sigma_Y }]]

[eqn: -1 ``<=`` rho ``<=`` 1]

■ Autocorrelation Function of the Random Process [eqn: X(t)]

[eqn: R_XX (t_1 , t_2)``=``E[X(t_1 ) X(t_2 ) ]]

[eqn: tau ``=``t_2 `-`t_1]

[eqn: R_XX (t_1 , t_1 +tau )``=``E[X(t_1 )X(t_1 + tau)]]

■ WSS (Wide Sense Stationary) Random Process

(1) [eqn: E[X(t)]``=``bar X ``=] constant

(2) [eqn: E[X(t)X(t+tau)]``=``R_XX (tau)]

■ Gaussian Random Process [eqn: X(t)]

Consider random process [eqn: X(t)] and define [eqn: N] random variables [eqn: X_1 `=`X(t_1 ), ```... ```, X_i `=`X(t_i ), ```... ```, X_N `=`X(t_N )] corresponding to [eqn: N] time instants [eqn: t_1 , ```...```, t_i , ```...```, t_N]. If, for any[eqn: N]=1, 2, ... and any times [eqn: t_1, ```...```, t_N], these random variables are jointly gaussian, the process is called gaussian.

■ Ergodic Process

[eqn: bar x ``=`` bar X]

Rxx[eqn: (tau)``=``R_XX (tau)]

the time average [eqn: bar x] and time autocorrelation Rxx[eqn: (tau)] equal the statistical averages [eqn: bar X] and statistical autocorrelation [eqn: R_XX (tau)] respectively.