Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PROBABILITY

A biased die gives probabilities^12 p,p,p,p,p, 2 pof throwing1, 2, 3, 4, 5, 6respectively.

If the random variableXis the number shown on the die and the random variableYis

defined asX^2 , calculate the covariance and correlation ofXandY.

We have already calculated in subsections 30.2.1 and 30.5.4 that

p=

2

13

,E[X]=

53

13

,E

[

X^2

]

=

253

13

,V[X]=

480

169

.

Using (30.135), we obtain

Cov[X, Y]=Cov[X, X^2 ]=E[X^3 ]−E[X]E[X^2 ].

NowE[X^3 ]isgivenby

E[X^3 ]=1^3 ×^12 p+(2^3 +3^3 +4^3 +5^3 )p+6^3 × 2 p

=

1313

2

p= 101,

and the covariance ofXandYis given by

Cov[X, Y] = 101−

53

13

×

253

13

=

3660

169

.

The correlation is defined by Corr[X, Y]=Cov[X, Y]/σXσY. The standard deviation of
Ymay be calculated from the definition of the variance. LettingμY=E[X^2 ]=^25313 gives

σ^2 Y=

p

2

(

12 −μY

) 2

+p

(

22 −μY

) 2

+p

(

32 −μY

) 2

+p

(

42 −μY

) 2

+p

(

52 −μY

) 2

+2p

(

62 −μY

) 2

=

187 356

169

p=

28 824

169

.

We deduce that

Corr[X, Y]=

3660

169

√

169

28 824

√

169

480

≈ 0. 984.

Thus the random variablesXandYdisplay a strong degree of positive correlation, as we
would expect.

We note that the covariance ofXandYoccurs in various expressions. For

example, ifXandYarenotindependent then

V[X+Y]=E

[

(X+Y)^2

]

−(E[X+Y])^2

=E

[

X^2

]

+2E[XY]+E

[

Y^2

]

−{(E[X])^2 +2E[X]E[Y]+(E[Y])^2 }

=V[X]+V[Y]+2(E[XY]−E[X]E[Y])

=V[X]+V[Y]+2 Cov[X, Y].