Page 238 - Fundamentals of Probability and Statistics for Engineers

Some Important Continuous Distributions 221

The pdf f (x) in Equation (7.67) is plotted in Figure 7.12 for several values
X
of n. It is shown that, as n increases, the shape of f (x) becomes more
X
symmetric. In view of Equation (7.68), since X can be expressed as a sum of
identically distributed random variables, we expect that the 2 distribution
approaches a normal distribution as n !1
on the basis of the central limit
theorem.
The mean and variance of random variable X having a 2 distribution are
easily obtained from Equation (7.57) as

2
m X ˆ n; ˆ 2n: 7:69†
X

7.5 BETA AND RELATED DISTRIBUTIONS

Whereas the lognormal and gamma distributions provide a diversity of one-
sided probability distributions, the beta distribution is rich in providing varied
probability distributions over a finite interval. The beta distribution is char-
acterized by the density function

8
‡ †
> 1 1
< x 1 x† ; for 0 x 1;
f x†ˆ † † 7:70†
X
>
0; elsewhere;
:
where parameters and take only positive values. The coefficient of f (x),
X
" ‡ )=‰ " ) " )Š; can be represented by 1/[B" , )], where
† †
B ; †ˆ ; 7:71†
‡ †

is known as the beta function, hence the name for the distribution given by
Equation (7.70).
The parameters and are both shape parameters; different combinations
of their values permit the density function to take on a wide variety of shapes.
When , > 1, the distribution is unimodal, with its peak at x ˆ " 1)/
" ‡ 2). It becomes U-shaped when , < 1; it is J-shaped when 1
and < 1; and it takes the shape of an inverted J when < 1 and 1.
Finally, as a special case, the uniform distribution over interval (0,1) results
when ˆ ˆ 1. Some of these possible shapes are displayed in Figures 7.13(a)
and 7.13(b).

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