∞
−n
− µ)
2
2n ( )
1
⎛ (x
⎞
L = C m x
exp −
2
dx
n3
s
∫
σ 2π
⎝ ⎜
2σ ⎠ ⎟
−∞
∞
1
Π( (n − + k)
1
(x − µ)
2
⎞
4
1
⎛
n
k
n
k
2
− −k
≈ C m
(−1)
(m)
(x − m)
exp −
2
dx
s
⎟ ⎟
∫ ∑ Πk
Π(n − 1)
σ 2π
⎝ ⎜
2σ ⎠
k =0
−∞
⎧
n n
( + 1)
⎫
−1
−2
2
2
− nm E X) − m) +
m E X ) − 2mE X) + m )
1
( (
( (
(
⎪
⎪
2
⎪
⎪
⎪
n n
( + 1)( (n + 2)
⎪
−3
3
2
2
3
−
m ( ( ) − 3 ( ) + 3m E X
( ) − m )
E X
mE X
n
⎪ ⎪
⎪ ⎪
6
= C m
s
⎨
⎬
⎪
⎪
( + )(n + 2)(n + 3)
n n 1
−
4
3
⎪ +
m E X ) − 4mE X )
4 ( ( (
(
⎪ ⎪
24
⎪
⎪
⎪
⎪
2
2
3
4
( + m )
⎪ + 6m E X
( ) − 4m E X)
⎩
⎭ ⎪
Therefore, the expected quality loss in the case of the normal distribution
of quality characteristic is of the following form:
⎧
⎫
2 n−2
2
2
⎪ n m
E X ) − 2 ( ) + m )
⎪
( (
mE X
⎪ ⎪
⎪
⎪ 2 n−3
3
2
2
3
⎪
L n = C s ⎨ −n m
E X ) − 3 ( ) + 3m E X
⎬
( (
mE X
( ) − m m )
⎪
⎪
⎪ (n
4
+ 11n
2 )
⎪
n− 4
4
3
2
2
3
4
m
E X ) − 4 ( ) + 6m E X ) − 4
( ) + m )
+
( (
mE X
(
m E X
⎪
⎪
⎩
12
⎭
(A2.20)
where
(
∞
1
⎛ (x − µ)
2
⎞
E X
n ) = x
n
exp −
2
dx
∫ σ
⎝ ⎜
2σ ⎠ ⎟
−∞
E X
0 ) 1
(
=
E X
1 ) µ
(
=
2
2 2
2
E X ) µ + σ
(
=
3
3
2
E X ) = µ + 3µσ
(
4
4
2 2
4
( ) = µ + 6µ σ + 3
E X
σ
2π
348 Appendix 2: Product Upgrades Based on Minimum Expected Quality Loss