Page 93 - Fundamentals of Probability and Statistics for Engineers

76 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

Efcgˆ c; 9
>
>
>
Efcg X†g ˆ cEfg X†g; >
=
4:3†
Efg X†‡ h X†g ˆ Efg X†g ‡ Efh X†g; >
>
>
>
Efg X†g Efh X†g; if g X† h X† for all values of X:
;
These relations follow directly from the definition of Efg X)g. For example,
Z 1
Efg X†‡ h X†g ˆ ‰g x†‡ h x†Šf x†dx
X
1
Z 1 Z 1
ˆ g x†f x†dx ‡ h x†f x†dx
X X
1 1
ˆ Efg X†g ‡ Efh X†g;
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

n
n
f
Let g(X ) ˆ X , n ˆ 1, 2, . . .; the expectation E X g , when it exists, is called the
nth moment of X. It is denoted by n and is given by
n
n
n ˆ EfX gˆ X x p x i †; for X discrete; 4:4†
i X
i
Z 1
n
n
n ˆ EfX gˆ x f x†dx; for X continuous: 4:5†
X
1
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 first moment of X is synonymously called the mean, expecta-
tion, or average value of X. A common notation for it is m X or simply m.
TLFeBOOK