fundamentals of engineering supplied-reference handbook - Ventech!

PROBABILITY AND DENSITY FUNCTIONS: MEANS AND VARIANCES
Variable Equation Mean Variance
Binomial
⎛n
⎞ n!
⎜ ⎟ =
Coefficient ⎝ x⎠
x!
( n − x)!
⎛n
⎞ x n−
x
Binomial b(
x;
n,
p)
= ⎜ ⎟ p ( 1 − p)
np np(1 – p)
⎝ x⎠
Hyper
Geometric
h
( x;
n,
r,
N )
Poisson f ( x;
)
⎛ N − r⎞
⎜ ⎟
⎛r
⎞ ⎝n
− x
=
⎠
⎜ ⎟
⎝ x⎠
⎛ N ⎞
⎜ ⎟
⎝n
⎠
x −λ
197
INDUSTRIAL ENGINEERING (continued)
r
( N − r)
n(
N − n)
2
N ( N −1)
λ e
λ =
λ λ
x!
Geometric g(x; p) = p (1 – p) x–1 1/p (1 – p)/p 2
Negative
Binomial
⎛ y + r −1⎞
f ⎜ ⎟ −
⎝ r −1
⎠
( ) ( ) y
r
y;
r,
p = p 1 p
r/p r (1 – p)/p 2
n!
x1
xk
Multinomial f ( x1
, …xk
) = p1
… pk
npi npi (1 – pi)
x ! , …,
x !
1
k
Uniform f(x) = 1/(b – a) (a + b)/2 (b – a) 2 /12
Gamma
f
( x)
α−1
−x
β
x
= α
β
e
Γ
( α)
−x
β
Exponential ( )
;
1
f x = e
β
α
β
α−1
−x
β
Weibull f ( x)
= x e
Normal ( x)
Triangular
f
( x)
α > 0,
β > 0
α
2
1 ⎛ x−µ
⎞
− ⎜ ⎟
2⎝
σ ⎠
β α 1
Γ
αβ αβ 2
β β 2
[ ( α + 1)
α]
β α 2
⎡ ⎛ α +
⎢Γ⎜
⎣ ⎝ α
⎞ ⎛ α + 1⎞⎤
⎟ − Γ ⎜ ⎟⎥
⎠ ⎝ α ⎠⎦
1 2
1
f = e
µ σ
σ 2π
2
⎧
⎪
= ⎨
⎪
⎪⎩
2(
x − a)
( b − a)(
m − a)
2(
b − x)
( b − a)(
b − m)
if a ≤ x ≤ m
if m < x ≤ b
nr
N
a + b + m
3
a
2
+ b
2
+ m
2
− ab − am − bm
18