Fundamentals of Probability and Statistics for Engineers

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 10,r 1, and

H ence, and

The desired probability is

According to Equation (7.80), the value of FX(050) can be found from

where Y is binomial and k 1 8, n 2 9, and p 0 50.
Using Table A.1, we find that

Equations (7.81) and (7.82) yield

7.5.2 G eneralized Beta D istribution

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

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

Some Important Continuous Distributions 225

ˆˆ

sˆ1.

ˆ 10  1  1 ‡ 1 ˆ9, ˆ 1 ‡ 1 ˆ2,

fX…x†ˆ

… 11 †

… 9 †… 2 †

x^8 … 1 x†;

ˆ

10!

8!

x^8 … 1 x†; for 0x 1 ;

ˆ 0 ; elsewhere:

P…X> 0 : 50 †ˆ 1 P…X 0 : 50 †ˆ 1 FX… 0 : 50 †: … 7 : 81 †

:

FX… 0 : 50 †ˆ 1 FY…k†; … 7 : 82 †

ˆ ˆˆ ‡ ˆˆ :

FY… 8 †ˆ 1 pY… 9 †ˆ 1  0 : 002 ˆ 0 : 998 : … 7 : 83 †

P…X> 0 : 50 †ˆ 1 FX… 0 : 50 †ˆ 1  1 ‡FY… 8 †ˆ 0 : 998 : … 7 : 84 †

Yˆ…ba†X‡a; … 7 : 85 †