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692
Probability and Random Variables
Appendix B
B–6 ENSEMBLE AVERAGE AND MOMENTS
Ensemble Average
One of the primary uses of probability theory is to evaluate the average value of a random
variable (which represents some physical phenomenon) or to evaluate the average value of
some function of the random variable. In general, let the function of the random variable be
denoted by y = h(x).
DEFINITION.
is given by
The expected value, which is also called the ensemble average, of y = h(x)
(B–21)
This definition may be used for discrete, as well as continuous, random variables. Note
that the operator is linear. The operator is
(B–22)
Note that other authors may denote the ensemble average of y by E [y] or 8y 9. We will use the
yq notation, since it is easier to write and more convenient to use when long equations are
being evaluated with a large number of averaging operators.
THEOREM. If x is a discretely distributed random variable, the expected value can be
evaluated by using
where M is the number of discrete points in the distribution.
Proof.
q
y = [h(x)] ! [h(x)] f(x) dx
L
[# ] =
L
q
y = [h(x)] = a
M
Using Eqs. (B–19) in (B–21), we get
q M
[h(x)] = h(x) c a P(x i ) d(x - x i ) d dx
L
- q
M q
= a P(x i ) h(x)d(x - x i ) dx
L
i=1
M
= a P(x i )h(x i )
i=1
- q
- q
i=1
-q
[# ] f(x) dx
i=1
h(x i )P(x i )
(B–23)
Example B–5 EVALUATION OF AN AVERAGE
We will now show that Eq. (B–23) and, consequently, the definition of expected value as given
by Eq. (B–21) are consistent with the way in which we usually evaluate averages. Suppose that
we have a class of n = 40 students who take a test. The resulting test scores are l paper with a