Page 241 - Fundamentals of Probability and Statistics for Engineers

224 Fundamentals of Probability and Statistics for Engineers

;
I x ( , ). If we write F X (x) with parameters and in the form F( ), the
x ,

correspondence between I x ( , ) and F( , x; , ) is determined as follows. If
, ,then
F x; ; †ˆ I x ; †: 7:75†
If < , then

F x; ; †ˆ 1 I 1 x† ; †: 7:76†

Another method of evaluating F X (x) in Equation (7.74) is to note the
similarity in form between f (x) and p (k) of a binomial random variable Y
X
Y
for the case where and are positive integers. We see from Equation
(6.2) that
n! n k
k
p k†ˆ p 1 p† ; k ˆ 0; 1; ... ; n: 7:77†
Y
k! n k†!
Also, f (x) in Equation (7.70) with and being positive integers takes the
X
form
‡ 1†! 1 1
f x†ˆ x 1 x† ; ; ˆ 1; 2; ... ; 0 x 1; 7:78†
X
1†! 1†!
and we easily establish the relationship

f x†ˆ ‡ 1†p k†; ; ˆ 1; 2; ... ; 0 x 1; 7:79†
X Y

where p (k) is evaluated at k ˆ 1, with n ˆ ‡ 2, and p ˆ x. For
Y
example, the value of f (0 5) with: ˆ 2, and ˆ 1, is numerically equal to
X
2p (1) with n ˆ 1, and p ˆ :
0 5; here p (1) can be found from Equation (7.77)
Y Y
or from Table A.1 for binomial random variables.
Similarly, the relationship between F X (x) and F Y (k) can be established. It
takes the form
F X x†ˆ 1 F Y k†; ; ˆ 1; 2; ... ; 0 x 1; 7:80†

with k ˆ 1, n ˆ ‡ 2, and p ˆ x. The PDF F Y (y) for a binomial
random variable Y is also widely tabulated and it can be used to advantage
here for evaluating F X (x) associated with the beta distribution.

Example 7.8. Problem: in order to establish quality limits for a manufactured
item, 10 independent samples are taken at random and the quality limits are

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