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

Hence, Q(m) = 1 − P (m) has the same properties as that of a distribution function, even
if we do not regard m as random variable in the present context. For a given (ρ 1 , ρ 2 , ..., ρ r )
the OC distribution for a C kind plan is given by 1 − P (m).
2.4.3 OC moments
Theorem 2.4.1
For the OC distribution defined as above,
E ( m k+1) = (k + 1)S [(x + k) (k) t(x 1 , ..., x r−1 , x, ρ 1 , ..., ρ r−1 )]
t ( ) denotes the individual term of the multinomial distribution viz.
t(x 1 , x 2 , ..., x r−1 , x, ρ 1 , ρ 2 , ..., ρ r−1 ) =
x!
x 1 !x 2 !...x r !(x − x r−1 )! ρx 1
1 ρ x 2
2 ...ρ x r−1
r−1 (1 − ρ (r−1) ) x−x (r−1)
)
and (x + k) (k) = (x + k)(x + k − 1)...(x + 1).
....(2.4.3)
We use here the symbol S for the summation with respect to x 1 , x 2 , ..., x r−1 and x over
the domain indicated in (2.4.2).
Proof
Let γ(m, α, β) = βe −mβ (mβ) α−1 /Γ(α), 0 ≤ m < ∞, α > 0, β > 0
then,
g(x, m) = γ(m, x + 1, 1)
−g ′ (x, m) = γ(m, x + 1, 1) − γ(m, x, 1).
Therefore, −P ′ (m) = S t(x 1 , ..., x r−1 , x, ρ 1 , ..., ρ r−1 )g ′ (x, m)
= S t(x 1 , ..., x r−1 , x, ρ 1 , ..., ρ r−1 )[γ(m, x + 1, 1) − γ(m, x, 1)]
and using the fact
we get (2.4.3).
∫ ∞
0
(m k )γ(m, x, 1) = x(x + 1)...(x + k − 1),
....(2.4.4)
2.4.4 The moment equivalent plans
From the above results we may now construct a moment equivalent single sampling plan
with parameter (a 0 , n 0 ) for a given vector of (ρ 1 , ρ 2 , ..., ρ r ) such that the single sampling plan
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