Page 240 - Fundamentals of Probability and Statistics for Engineers

Some Important Continuous Distributions 223

The mean and variance of a beta-distributed random variable X are, follow-
ing straightforward integrations,

9
m X ˆ ; >
‡ >
=
7:72†
2
ˆ 2 : >
>
X
‡ † ‡ ‡ 1† ;
Because of its versatility as a distribution over a finite interval, the beta
distribution is used to represent a large number of physical quantities for which
values are restricted to an identifiable interval. Some of the areas of application
are tolerance limits, quality control, and reliability.
An interesting situation in which the beta distribution arises is as follows.
Suppose a random phenomenon Y can be observed independently n times and,
after these n independent observations are ranked in order of increasing mag-
nitude, let y r and y n s‡1 be the values of the rth smallest and sth largest
observations, respectively. If random variable X is used to denote the propor-
tion of the original Y taking values between y r and y n s‡1 , it can be shown that
X follows a beta distribution with ˆ n r s ‡ 1, and ˆ r ‡ s; that is.

n ‡ 1† r‡s 1
8
> n r s 1 x† ; for 0 x 1;
< x
f x†ˆ n r s ‡ 1† r ‡ s† 7:73†
X
>
0; elsewhere:
:
This result can be found in Wilks (1942). We will not prove this result but we
will use it in the next section, in Example 7.8.

7.5.1 PROBABILITY TABULATIONS

The probability distribution function associated with the beta distribution is

0; for x < 0;
8
>
>
>
‡ † 1 1
> Z x
<
F X x†ˆ u 1 u† du; for 0 x 1; 7:74†
† †
> 0
>
>
>
:
1; for x > 1;
which can be integrated directly. It also has the form of an incomplete beta
function for which values for given values of and can be found from
mathematical tables. The incomplete beta function is usually denoted by

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