Fundamentals of Probability and Statistics for Engineers

(John Hannent) #1

Ix( , ). If we write FX(x) with parameters and in the form F( ), the
correspondence between Ix( , ) and F( ) is determined as follows. If
,then


If then


Another method of evaluating FX(x) in Equation (7.74) is to note the
similarity in form between fX(x) and pY (k) of a binomial random variable Y
for the case where and are positive integers. We see from Equation
(6.2) that


Also, fX (x) in Equation (7.70) with and being positive integers takes the
form


and we easily establish the relationship


where pY (k) is evaluated at with and For
example, the value of fX (05) with and is numerically equal to
2 pY (1) with n 1, and p 0 5; here pY (1) can be found from Equation (7.77)
or from Table A.1 for binomial random variables.
Similarly, the relationship between FX(x) and FY (k) can be established. It
takes the form


with and The PDF FY (y) for a binomial
random variable Y is also widely tabulated and it can be used to advantage
here for evaluating FX(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


224 Fundamentals of Probability and Statistics for Engineers


x; ,
, x; ,
 ,


F…x;;†ˆIx…;†: … 7 : 75 †

<,

F…x;;†ˆ 1 I… 1 x†…;†: … 7 : 76 †

pY…k†ˆ

n!
k!…nk†!

pk… 1 p†nk; kˆ 0 ; 1 ;...;n: … 7 : 77 †

fX…x†ˆ

… ‡  1 †!

…  1 †!…  1 †!

x^ ^1 … 1 x†^ ^1 ;;ˆ 1 ; 2 ;...; 0 x 1 ;… 7 : 78 †

fX…x†ˆ… ‡  1 †pY…k†;;ˆ 1 ; 2 ;...; 0 x 1 ; … 7 : 79 †

kˆ 1, nˆ ‡ 2, pˆx.
: ˆ2, ˆ1,
ˆˆ :

FX…x†ˆ 1 FY…k†;;ˆ 1 ; 2 ;...; 0 x 1 ; … 7 : 80 †

kˆ 1,nˆ ‡ 2, pˆx.
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