Page 104 - Fundamentals of Probability and Statistics for Engineers

Expectations and Moments 87

or
3
P 2:94 X 3:06† :
4

In words, the probability of a three-foot tape measure being in error less than
or equal to :
0 06 feet is at least 0.75. Various probability bounds can be found
by assigning different values to k.
The complete generality with which the Chebyshev inequality is derived
suggests that the bounds given by Equation (4.17) can be quite conservative.
This is indeed true. Sharper bounds can be achieved if more is known about the
distribution.

4.3 MOMENTS OF TWO OR MORE RANDOM VARIABLES
Let g(X , Y ) be a real-valued function of two random variables X and Y . Its
expectation is defined by

X X
Efg X; Y†g ˆ g x i ; y j †p x i ; y j †; X and Y discrete; 4:18†
XY
i j
Z 1 Z 1
Efg X; Y†g ˆ g x; y†f x; y†dxdy; X and Y continuous; 4:19†
XY
1 1
if the indicated sums or integrals exist.
In a completely analogous way, the joint moments nm of X and Y are given
by, if they exist,
n
m
nm ˆ EfX Y g: 4:20†
m
n
They are computed from Equation (4.18) or (4.19) by letting g(X , Y ) ˆ X Y .
Similarly, the joint central moments of X and Y , when they exist, are
given by
n m
nm ˆ Ef X m X † Y m Y † g: 4:21†
They are computed from Equation (4.18) or (4.19) by letting

m
n
g X; Y†ˆ X m X † Y m Y † :
Some of the most important moments in the two-random-variable case are
clearly the individual means and variances of X and Y . In the notation used

TLFeBOOK