The pdf fX (x) in Equation (7.67) is plotted in Figure 7.12 for several values
of n. It is shown that, as n increases, the shape of fX(x) becomes more
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 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
7.5 Beta and R elated D istributions
Whereas the lognormal and gamma distributions provide a diversity of one-
sided probability distributions, the beta distribution is rich inproviding varied
probability distributions over a finite interval. The beta distribution is char-
acterized by the density function
where parameters and take only positive values. The coefficient of fX(x),
can be represented by where
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 the distribution is unimodal, with its peak at
It becomes U-shaped when it is J-shaped when
and and it takes the shape of an inverted J when and
Finally, as a special case, the uniform distribution over interval (0,1) results
when 1. Some of these possible shapes are displayed in Figures 7.13(a)
and 7.13(b).
Some Important Continuous Distributions 221
!1
mXn;^2 X 2 n:
7 : 69 fX
x
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>:
7 : 70
" )=" )" ); 1/[B" , )],
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; 7 : 71
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