Introduction to Probability and Statistics for Engineers and Scientists

126 Chapter 4:Random Variables and Expectation

⇔

P{X=1,Y= 1 }

P{X= 1 }

>P{Y = 1 }

⇔P{Y= 1 |X= 1 }>P{Y= 1 }

That is, the covariance ofXandYis positive if the outcomeX=1 makes it more likely
thatY=1 (which, as is easily seen by symmetry, also implies the reverse).
In general, it can be shown that a positive value of Cov(X,Y) is an indication that
Ytends to increase asXdoes, whereas a negative value indicates thatYtends to decrease
asXincreases. The strength of the relationship betweenX andY is indicated by the
correlation betweenXandY, a dimensionless quantity obtained by dividing the covari-
ance by the product of the standard deviations ofXandY. That is,

Corr(X,Y)=

Cov(X,Y)

√

Var(X)Var(Y)

It can be shown (see Problem 49) that this quantity always has a value between−1 and+1.

4.8Moment Generating Functions

The moment generating functionφ(t) of the random variableXis defined for all values
tby

φ(t)=E[etX]=







∑

x

etxp(x)ifXis discrete

∫∞

−∞

etxf(x)dx ifXis continuous

We callφ(t) the moment generating function because all of the moments ofXcan be
obtained by successively differentiatingφ(t). For example,

φ′(t)=

d

dt

E[etX]

=E

[

d

dt

(etX)

]

=E[XetX]

Hence,

φ′(0)=E[X]

Similarly,

φ′′(t)=

d

dt

φ′(t)

=

d

dt

E[XetX]