30.12 PROPERTIES OF JOINT DISTRIBUTIONS
One particularly useful consequence of its definition is that the covarianceof twoindependentvariables,XandY, is zero. It immediately follows from
(30.134) that their correlation is also zero, and this justifies the use of the term
‘uncorrelated’ for two such variables. To show this extremely important property
we first note that
Cov[X, Y]=E[(X−μX)(Y−μY)]=E[XY−μXY−μYX+μXμY]=E[XY]−μXE[Y]−μYE[X]+μXμY
=E[XY]−μXμY. (30.135)Now, ifXandY are independent thenE[XY]=E[X]E[Y]=μXμY and so
Cov[X, Y] = 0. It is important to note that the converse of this result is not
necessarily true; two variables dependent on each other can still be uncorrelated.
In other words, it is possible (and not uncommon) for two variablesXandY
to be described by a joint distributionf(x, y)thatcannotbe factorised into a
product of the formg(x)h(y), but for which Corr[X, Y] = 0. Indeed, from the
definition (30.133), we see that for any joint distributionf(x, y) that is symmetric
inxaboutμX(or similarly iny) we have Corr[X, Y]=0.
We have already asserted that if the correlation of two random variables ispositive (negative) they are said to be positively (negatively) correlated. We have
also stated that the correlation lies between−1 and +1. The terminology suggests
that if the two RVs are identical (i.e.X=Y) then they are completely correlated
and that their correlation should be +1. Likewise, ifX=−Ythen the functions
are completely anticorrelated and their correlation should be−1. Values of the
correlation function between these extremes show the existence of some degree
of correlation. In fact it is not necessary thatX=Yfor Corr[X, Y] = 1; it is
sufficient thatYis a linear function ofX,i.e.Y=aX+b(withapositive). Ifa
is negative then Corr[X, Y]=−1. To show this we first note thatμY=aμX+b.
Now
Y=aX+b=aX+μY−aμX ⇒ Y−μY=a(X−μX),and so using the definition of the covariance (30.133)
Cov[X, Y]=aE[(X−μX)^2 ]=aσX^2.It follows from the properties of the variance (subsection 30.5.3) thatσY=|a|σX
and so, using the definition (30.134) of the correlation,
Corr[X, Y]=aσ^2 X
|a|σ^2 X=a
|a|,which is the stated result.
It should be noted that, even if the possibilities ofXandYbeing non-zero aremutually exclusive, Corr[X, Y] need not have value±1.