Page 35 - Computational Statistics Handbook with MATLAB

Chapter 2: Probability Concepts 21

BAYES’ THEOREM

(
()PF E i )
PE i F( ) = ----------------------------------------------------------------------------------------- . (2.13)
PE i
(
(
(
(
PE 1 )PF E 1 ) + … + PE k )PF E k )
2.4 Expectation
Expected values and variances are important concepts in statistics. They are
used to describe distributions, to evaluate the performance of estimators, to
obtain test statistics in hypothesis testing, and many other applications.

Variance
Mea
Me anna andndV ariance
aann
aandnd
VVarianceariance
MeMe
The mean or expected value of a random variable is defined using the proba-
bility density (mass) function. It provides a measure of central tendency of
the distribution. If we observe many values of the random variable and take
the average of them, we would expect that value to be close to the mean. The
expected value is defined below for the discrete case.
EXPECTED VALUE - DISCRETE RANDOM VARIABLES
∞
∑ () . (2.14)
µ = EX[] = x i fx i
i = 1
We see from the definition that the expected value is a sum of all possible
values of the random variable where each one is weighted by the probability
that X will take on that value.
The variance of a discrete random variable is given by the following defi-
nition.

VARIANCE - DISCRETE RANDOM VARIABLES
For µ < ∞ ,

∞
()
( [
2
2
2
σ = VX() = EX – µ) ] = ∑ ( x i – µ) fx i . (2.15)
i = 1

© 2002 by Chapman & Hall/CRC