Multiattribute acceptance sampling plans - Library(ISI Kolkata ...

...(1.1.2)
If now X i is assumed to follow binomial with parameters N and p i and they are independent
as given by (1.1.1), the unconditional probability of obtaining (x 1 , x 2 , ..., x r ) defectives in
sample of size n is
∑ ∑
... ∑
X 1 X 2 X r
r∏
i=1
[( )( ) ( )] [( )
]
n N − n N N
/
p X i
i (1 − p i ) N−X i
x i X i − x i X i X i
( )
r∏ n x
= p i i (1 − p i ) n−x i
.
x i
i=1
If b(., ., . ) denotes an individual term of the binomial distribution, then
r∏
P r(x 1 , x 2 ..., x r | p 1 , p 2 ...p r ) = b(x i , n i , p i ), x i = 0, 1, 2, ..., n; 0 < p i < 1; i = 1, 2, ..., r.
i=1
...(1.1.3)
b) When the defect occurrences are mutually exclusive
When the defect occurrences are mutually exclusive, the expression for the probability of
observing (x 1 , x 2 , ..., x r ) defective in a sample of size n from a lot of size N containing
(X 1 , X 2 , ..., X r ) defectives will be multivariate hypergeometric as
P r(x 1 , x 2 , ..., x r | X 1 , X 2 , ..., X r ) =
( )( ) ( )( ) ( )
X1 X2 Xr N − X1 − X 2 ... − X r N
...
/
x 1 x 2 x r n − x 1 − x 2 ... − x r n
...(1.1.4)
At any process average vector (p 1 , p 2 , ..., p r ) the joint probability distribution of (X 1 , X 2 , ..., X r )
can be assumed to be multinomial (N, p 1 , p 2 , ..., p r ) such that
( )( ) ( )
N N − X(1) N − X(r−1)
P r(X 1 , X 2 , ..., X r | p 1 , p 2 , ..., p r ) =
...
p X 1
1 ...p Xr
r (1−p (r) ) (N−X (r))
X 1 X 2 X r
where
X (i) = X 1 + X 2 + ... + X i and p (i) = p 1 + p 2 + ... + p i ; i = 1, 2, ..., r.
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