Correlation in Random Variables

Correlation in Random VariablesLecture 11 Spring 2002 Correlation in Random VariablesSuppose that an experimentproduces two Random vari-ables, about the relationship be-tween them?One of the best ways to visu-alize the possible relationshipis to plot the (X, Y)pairthatis produced by several trials ofthe experiment. An exampleof correlated samples is shownat the rightLecture 111 Joint Density FunctionThe joint behavior ofXandYis fully captured in the joint probabilitydistribution. For a continuous distributionE[XmYn]= xmynfXY(x, y)dxdyFor discrete distributionsE[XmYn]= x Sx y SyxmynP(x, y)Lecture 112 Covariance FunctionThe covariance function is a number that measures the commonvariation is defined ascov(X, Y)=E[(X E[X])(Y E[Y])]=E[XY] E[X]E[Y]The covariance is determined by the difference inE[XY]andE[X]E[Y].IfXandYwere statistically independent thenE[XY] would equalE[X]E[Y] and the covariance would be covariance of a Random variable with itself is equal to its [X, X]=E[(X E[X])2]=var[X]Lecture 113 Correlation CoefficientThe covariance can be normalized to produce what is known as thecorrelation coefficient.

• A random process is a rule that maps every outcome e of an experiment to a function X(t,e). • A random process is usually conceived of as a function of time, but there is no reason to not consider random processes that are functions of other independent variables, such as spatial coordi-nates. • The function X(u,v,e) would be a function ...

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