Page 36 - Computational Statistics Handbook with MATLAB

22 Computational Statistics Handbook with MATLAB

From Equation 2.15, we see that the variance is the sum of the squared dis-
tances, each one weighted by the probability that X = x i . Variance is a mea-
sure of dispersion in the distribution. If a random variable has a large
variance, then an observed value of the random variable is more likely to be
σ
far from the mean µ. The standard deviation is the square root of the vari-
ance.
The mean and variance for continuous random variables are defined simi-
larly, with the summation replaced by an integral. The mean and variance of
a continuous random variable are given below.

EXPECTED VALUE - CONTINUOUS RANDOM VARIABLES

∞
µ = EX[] = ∫ xf x()d . x (2.16)

– ∞

VARIANCE - CONTINUOUS RANDOM VARIABLES
For µ < ∞ ,

∞
2 2 2
( [
σ = VX() = EX – µ) ] = ∫ ( x – µ) fx()d . x (2.17)
– ∞
We note that Equation 2.17 can also be written as

[
[
2
2
2
VX() = EX ] – µ = EX ] – ( EX[]) 2 .
Other expected values that are of interest in statistics are the moments of a
random variable. These are the expectation of powers of the random variable.
In general, we define the r-th moment as
[
µ' r = EX r , ] (2.18)
and the r-th central moment as

( [
µ
µ r = EX – ) r . ] (2.19)
.
The mean corresponds to µ' 1 and the variance is given by µ 2

© 2002 by Chapman & Hall/CRC