C. Relations of Probability Distributions

Appendix



D.



Probability Functions



Pascal distribution



Exponential distribution



Geometric distribution



Binomial distribution



Probability Probability density function

Distri

f(x)

bution



Failure Rate

λ(x)



Mean

Value

μ



Variance

σ2



np



Remark



npn



n

⎛n⎞

n

fBin (x, n, p) = ⎜ ⎟ px q n −x R Bin (x) = ∑ ⎛ ⎞ pi qn −i

⎜ ⎟

⎝x ⎠

i =x+1⎝ 1 ⎠

1.0



qn

0 1 2 3 4 .....



x



x



0 1 2 3 4 .....



n



f Geo (x) = qx p



n



p

λ Geo = q = const



R Geo(x) = qx + 1

0



1.0

p



q



0 1 2 3 4 .....



x



n



0



0 1 2 3 4 .....



n



x



0 1 2 3 4 .....



λ



n



1



1.0



λ

e



x



λ exp = λ = const



R exp (x) = e− λx



f exp (x) = λ e − λx



p2



1



0



q



p



q

p



1

e

x



0

0



1/ λ



f Pas(x) = ⎛

⎜

⎝



x + y −1⎞ x y

⎟p q

y ⎠



y=1



x



λ Pas =



α



β<1



RPas

y>1

y=1



β=1



0



x

0 1 2 3 4 .....



1.0

1 < β1

β2 (> β1)



x



p2



f Pas



y2 (> y1)

1 < y1

y=1



0 1 2 3 4 .....



RΓ (x) =

α ∞(αt) β−1e− αt dt

Γ (β) ∫ x



(x, α, β)



qy



p



n −1



p

0 1 2 3 4 .....



qy



0



RPas (x) = ∑ (n−1) pn− iq i

i =x i



y2 (> y1)



λ



x



x 0



0



1 < y1



2



λ

0



pq



0



Gamma distribution



Reliability function

R(x)



λΓ =



x



fΓ



α: Scale

palameter



RΓ



α



β>1



β

β>1



β=1

β<1



β=1



x



β<1



α



α



β: Shape

palameter



β

2



Refer to

Figure B.13



0



Rev. 1.01 Nov. 28, 2008 Page 409 of 410

REJ27L0001-0101



Appendix



1 ∞ n −t

t e dt

n! ∫ x



λ Pois =



n≥1



x



x2



1.0



fwbl(x) =

m

m(x− γ)m −1• e−(x− γ)

t0

t0



n≥1



x



0

−



R wbl(x) = e



(x− γ) m

t0



λ Wdl(x) =



m

(x− γ )m−1

t0



m>2



m<1

m>2



m>2



2>m>1



m = 2 1.0

−



e

γ



x



f Norm (x) = 1 − (x − μ)2

e 2σ 2

σ 2π



m=1



m<1



t0



0



γ



γ+1



R Norm (x) =

1 ∞−

σ 2π ∫ x e



1.0



m=2

2>m>1



1

t0



m=1



1



x



0



γ



γ+1



λ Norm =



(t− μ) 2

2σ 2 dt



m<1

x



m: Shape

palameter

γ: Position

palameter

t0: Scale

palameter

Refer to

Figure B.16



f Nom

R Norm



1.0



μ



1



α 2π



Refer to

Figure B.14



x



0



2

2

1

2

t0 m {Γ (1+ m ) −Γ (1+ m )}



x



Remark



f Pois



n=0



n



m=1



Variance

σ2



RPois



n=0



n≥1



1

t0



Mean

Value

μ



σ2



Refer to

Figure B.15



0.5



x



γ



0



f log - norm (x) =

1 −

σx 2π e



(lix− lnxo)2

2σ 2



0



γ



x



x

0



R log - norm (x) =

∞

∫x f log - norm(t)dt



λ log −norm =



fl − norm

Rl − norm



1.0



1

2



1

2

0



x



X0



Rev. 1.01 Nov. 28, 2008 Page 410 of 410

REJ27L0001-0101



x



0



x



2

e 2Inx0+σ2 × (eσ −1)



Weibull distribution



R pois (x) =

1.0



fpois



0



Normal distribution



n!



n −x



x e



n=0



1.0

(n)



0



Logarithmic normal

distribution



1



Failure Rate

λ(x)



1

1

m

t0 Γ (1+ m )



f pois (x) =



Reliability function

R(x)



σ2

e − (Inx0+ 2 )



Poisson distribution



Probability Probability density function

Distri

f(x)

bution



σ2: Variance

of normal

distribution

X0: median

value of

probability

distribution



Semiconductor Reliability Handbook

Publication Date: Rev.1.00, August 31, 2006

Rev.1.01, November 28, 2008

Published by:

Sales Strategic Planning Div.

Renesas Technology Corp.

Edited by:

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Global Strategic Communication Div.

Renesas Solutions Corp.

© 2008. Renesas Technology Corp., All rights reserved. Printed in Japan.



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