Fundamentals of Probability and Statistics for Engineers

Let us note some basic properties associated with the expectation operator.
For any constant c and any functions g(X) and h(X) for which expectations
exist, we have

These relations follow directly from the definition of F or example,

as given by the third of Equations (4.3). The proof is similar when X is discrete.

4.1 Moments of a Single Random Variable

Let g(X) Xn, n 1, 2,.. .; the expectation E Xn^ , when it exists, is called the
nth moment of X. It is denoted by n and is given by

4.1.1 Mean, Median, and Mode

One of the most important moments is 1 , the first moment. Using the mass
analogy for the probability distribution, the first moment may be regarded as
the center of mass of its distribution. It is thus the average value of random
variable X and certainly reveals one of the most important characteristics of its
distribution. The fir st moment of X is synonymously called the mean, expecta-
tion, or average value of X. A common notation for it is mX or simply m.

76 Fundamentals of Probability and Statistics for Engineers

Efcgˆc;

Efcg…X†gˆcEfg…X†g;

Efg…X†‡h…X†gˆEfg…X†g‡Efh…X†g;

Efg…X†gEfh…X†g; ifg…X†h…X†for all values ofX:

9

>>>

>=

>>

>>

;

… 4 : 3 †

EfgX)g.

Efg…X†‡h…X†gˆ

Z 1

1

‰g…x†‡h…x†ŠfX…x†dx

ˆ

Z 1

1

g…x†fX…x†dx‡

Z 1

1

h…x†fX…x†dx

ˆEfg…X†g‡Efh…X†g;

(^) nˆEfXngˆ

X

i

xnipX…xi†;forXdiscrete; … 4 : 4 †

(^) nˆEfXngˆ

Z 1

1

xnfX…x†dx;forXcontinuous: … 4 : 5 †

ˆˆfg