Fundamentals of Probability and Statistics for Engineers

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

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 in cr easing mag-
nitude, let yr and 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 yr and it can be shown that
X follows a beta distribution with 1, and that is.

This result can be found in Wilks(1942). We will notprove 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

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

Some Important Continuous Distributions 223

mXˆ

‡

;

^2 Xˆ

… ‡ †^2 … ‡ ‡ 1 †

:

9

>>=

>>

;

… 7 : 72 †

yns‡ 1

yns‡ 1 ,

ˆnrs‡ ˆr‡s;

fX…x†ˆ

…n‡ 1 †

…nrs‡ 1 †…r‡s†

xnrs… 1 x†r‡s^1 ; for 0x 1 ;

0 ; elsewhere:

8

><

>:

… 7 : 73 †

FX…x†ˆ

0 ; forx< 0 ;

… ‡ †

… †… †

Zx

0

u^ ^1 … 1 u†^ ^1 du; for 0x 1 ;

1 ; forx> 1 ;

8

>>

>>

<

>>

>>:

… 7 : 74 †