Page 242 - Fundamentals of Probability and Statistics for Engineers

Some Important Continuous Distributions 225

established by using the lowest and highest sample values. What is the prob-
ability that at least 50% of the manufactured items will fail within these limits?
Answer: let X be the proportion of items taking values within the established
limits. Its pdf thus takes the form of Equation (7.73), with n ˆ 10, r ˆ 1, and
s ˆ 1.
Hence, ˆ 10 1 1 ‡ 1 ˆ 9, ˆ 1 ‡ 1 ˆ 2, and

11†
8
f x†ˆ x 1 x†;
X
9† 2†
10!
8
ˆ x 1 x†; for 0 x 1;
8!
ˆ 0; elsewhere:
The desired probability is

P X > 0:50†ˆ 1 P X 0:50†ˆ 1 F X 0:50†: 7:81†
:
According to Equation (7.80), the value of F X (0 50) can be found from
F X 0:50†ˆ 1 F Y k†; 7:82†

where Y is binomial and k ˆ 1 ˆ 8, n ˆ ‡ 2 ˆ 9, and p ˆ :
0 50.
Using Table A.1, we find that
F Y 8†ˆ 1 p 9†ˆ 1 0:002 ˆ 0:998: 7:83†
Y

Equations (7.81) and (7.82) yield
P X > 0:50†ˆ 1 F X 0:50†ˆ 1 1 ‡ F Y 8†ˆ 0:998: 7:84†

7.5.2 GENERALIZED BETA DISTRIBUTION

The beta distribution can be easily generalized from one restricted to unit
interval (0, 1) to one covering an arbitrary interval (a, b). Let Y be such
a generalized beta random variable. It is clear that the desired transforma-
tion is

Y ˆ b a†X ‡ a; 7:85†

where X is beta-distributed according to Equation (7.70). Equation (7.85)
represents a monotonic transformation from X and Y and the procedure

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