Mathematical Methods for Physics and Engineering : A Comprehensive Guide

STATISTICS

31.4.6 Population covarianceCov[x, y]and correlationCorr[x, y]

So far we have assumed that each of ourNindependent samples consists of

a single numberxi. Let us now extend our discussion to a situation in which

each sample consists of two numbersxi,yi, which we may consider as being

drawn randomly from a two-dimensional populationP(x, y). In particular, we

now consider estimators for the population covariance Cov[x, y] and for the

correlation Corr[x, y].

Whenμxandμyareknown, an appropriate estimator of the population covari-

ance is

Cov[̂x, y]=xy−μxμy=

(

1

N

∑N

i=1

xiyi

)

−μxμy. (31.59)

This estimator is unbiased since

E

[

Cov[̂x, y]

]

=

1

N

E

[N

∑

i=1

xiyi

]

−μxμy=E[xiyi]−μxμy=Cov[x, y].

Alternatively, ifμxandμyareunknown, it is natural to replaceμxandμyin

(31.59) by the sample means ̄xand ̄yrespectively, in which case we recover the

sample covarianceVxy=xy− ̄x ̄ydiscussed in subsection 31.2.4. This estimator

is biased but an unbiased estimator of the population covariance is obtained by

forming

Cov[̂x, y]= N

N− 1

Vxy. (31.60)

Calculate the expectation value of the sample covarianceVxyfor a sample of sizeN.

The sample covariance is given by

Vxy=

(

1

N

∑

i

xiyi

)

−

(

1

N

∑

i

xi

)(

1

N

∑

j

yj

)

.

Thus its expectation value is given by

E[Vxy]=

1

N

E

[

∑

i

xiyi

]

−

1

N^2

E

[(

∑

i

xi

)(

∑

j

xj

)]

=E[xiyi]−

1

N^2

E







∑

i

xiyi+

∑

i,j

j=i

xiyj





