A Concise Summary of @RISK Probability Distribution Functions
Copyright © 2002 - 2005 Palisade Corporation All Rights Reserved
Beta RISKBeta(α1, α2)
Parameters:
α1 continuous shape parameter α1 > 0
α2 continuous shape parameter α2 > 0
Domain:
0 ≤ x ≤ 1 continuous
Density and Cumulative Distribution Functions:
xα1 −1()1− x α 2 −1 f (x) = Β(α1,α2 )
Bx ()α1,α2 F(x) = ≡ Ix ()α1,α2 B(α1,α2 ) where B is the Beta Function and Bx is the Incomplete Beta Function.
Mean:
α 1 α1 + α2
Variance:
α α 1 2 2 ()α1 + α2 (α1 + α2 +1)
Skewness:
α − α α + α +1 2 2 1 1 2 α1 + α2 + 2 α1α2
Kurtosis:
()α + α +1 2(α + α )2 + α α (α + α − 6) 3 1 2 ( 1 2 1 2 1 2 ) α1α2 ()α1 + α2 + 2 (α1 + α2 + 3)
Mode:
α1 −1 α1>1, α2>1 α1 + α2 − 2
0 α1<1, α2≥1 or α1=1, α2>1
1 α1≥1, α2<1 or α1>1, α2=1
PDF - Beta(2,3) CDF - Beta(2,3)
2.0 1.0
1.8
1.6 0.8
1.4
1.2 0.6
1.0
0.8 0.4
0.6
0.4 0.2
0.2
0.0 0.0 0 2 4 6 8 0 2 0 2 4 6 8 0 2 2 2 . . 0. 0. 0. 0. 0. 1. 1. 0. 0. 0. 0. 0. 1. 1. -0 -0
Beta (Generalized) RISKBetaGeneral(α1, α2, min, max)
Parameters:
α1 continuous shape parameter α1 > 0
α2 continuous shape parameter α2 > 0
min continuous boundary parameter min < max
max continuous boundary parameter
Domain:
min ≤ x ≤ max continuous
Density and Cumulative Distribution Functions:
()x − min α1 −1(max− x)α2 −1 f (x) = α1 +α2 −1 Β(α1,α2 )()max− min
Bz ()α1,α2 x − min F(x) = ≡ Iz (α1,α2 ) with z ≡ B(α1,α2 ) max− min where B is the Beta Function and Bz is the Incomplete Beta Function.
Mean:
α min+ 1 ()max− min α1 + α2
Variance: α α 1 2 ()max− min 2 2 ()α1 + α2 (α1 + α2 +1)
Skewness:
α − α α + α +1 2 2 1 1 2 α1 + α2 + 2 α1α2
Kurtosis:
()α + α +1 2(α + α )2 + α α (α + α − 6) 3 1 2 ( 1 2 1 2 1 2 ) α1α2 ()α1 + α2 + 2 (α1 + α2 + 3)
Mode:
α1 −1 min+ (max− min) α1>1, α2>1 α1 + α2 − 2
min α1<1, α2≥1 or α1=1, α2>1
max α1≥1, α2<1 or α1>1, α2=1
PDF - BetaGeneral(2,3,0,5) CDF - BetaGeneral(2,3,0,5)
0.40 1.0
0.35 0.8 0.30
0.25 0.6
0.20
0.4 0.15
0.10 0.2 0.05
0.00 0.0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 -1 -1
Beta (Subjective) RISKBetaSubj(min, m.likely, mean, max)
Definitions:
min+ max mid ≡ 2
()mean − min (mid − m.likely) max− mean α ≡ 2 α ≡ α 1 ()mean − m.likely (max− min)2 1 mean − min
Parameters:
min continuous boundary parameter min < max
m.likely continuous parameter min < m.likely < max
mean continuous parameter min < mean < max
max continuous boundary parameter
mean > mid if m.likely > mean mean < mid if m.likely < mean mean = mid if m.likely = mean
Domain: min ≤ x ≤ max continuous
Density and Cumulative Distribution Functions:
()x − min α1 −1(max− x)α2 −1 f (x) = α1 +α2 −1 Β(α1,α2 )()max− min
Bz ()α1,α2 x − min F(x) = ≡ Iz (α1,α2 ) with z ≡ B(α1,α2 ) max− min where B is the Beta Function and Bz is the Incomplete Beta Function.
Mean:
mean
Variance:
()mean − min (max− mean)(mean − m.likely)
2 ⋅ mid + mean − 3⋅ m.likely
Skewness:
2()mid − mean mean − m.likely 2 ⋅ mid + mean − 3⋅ m.likely ()( ) mean + mid − 2 ⋅ m.likely ()mean − min (max− mean)
Kurtosis:
()α + α +1 2(α + α )2 + α α (α + α − 6) 3 1 2 ( 1 2 1 2 1 2 ) α1α2 ()α1 + α2 + 2 (α1 + α2 + 3)
Mode:
m.likely
PDF - BetaSubj(0,1,2,5) CDF - BetaSubj(0,1,2,5)
0.30 1.0
0.25 0.8
0.20 0.6
0.15
0.4 0.10
0.2 0.05
0.00 0.0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 -1 -1
Binomial RISKBinomial(n, p)
Parameters:
n discrete “count” parameter n > 0 *
p continuous “success” probability 0 < p < 1 *
*n = 0, p = 0 and p = 1 are supported for modeling convenience, but give degenerate distributions.
Domain:
0 ≤ x ≤ n discrete integers
Mass and Cumulative Functions:
⎛n ⎞ x n−x f (x) = ⎜ ⎟p (1− p) ⎝ x⎠
x ⎛n⎞ F(x) = ⎜ ⎟ pi (1− p)n −i ∑⎜ i ⎟ i=0 ⎝ ⎠
Mean:
np
Variance:
np()1− p
Skewness:
()1− 2p
np()1− p
Kurtosis:
6 1 3 − + n np(1− p)
Mode:
(bimodal) p(n +1)−1 and p(n +1) if p(n +1) is integral
(unimodal) largest integer less than p(n +1) otherwise
PMF - Binomial(8,.4) CDF - Binomial(8,.4)
0.30 1.0
0.25 0.8
0.20 0.6
0.15
0.4 0.10
0.2 0.05
0.00 0.0 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 -1 -1
Chi-Squared RISKChiSq(ν)
Parameters:
ν discrete shape parameter ν > 0
Domain:
0 ≤ x < +∞ continuous
Density and Cumulative Functions:
1 f (x) = e−x 2x (ν 2)−1 2ν 2 Γ()ν 2
Γx 2 ()ν 2 F(x) = Γ()ν 2
where Γ is the Gamma Function, and Γx is the Incomplete Gamma Function.
Mean:
ν
Variance:
2ν
Skewness:
8
ν
Kurtosis:
12 3 + ν
Mode:
ν-2 if ν ≥ 2
0 if ν = 1
PDF - ChiSq(5) CDF - ChiSq(5)
0.18 1.0
0.16 0.9
0.8 0.14 0.7 0.12 0.6 0.10 0.5 0.08 0.4 0.06 0.3 0.04 0.2
0.02 0.1
0.00 0.0 0 2 4 6 8 0 2 4 6 8 -2 -2 10 12 14 16 10 12 14 16
Cumulative (Ascending) RISKCumul(min, max, {x}, {p})
Parameters:
min continuous parameter min < max
max continuous parameter
{x} = {x1, x2, …, xN} array of continuous parameters min ≤ xi ≤ max
{p} = {p1, p2, …, pN} array of continuous parameters 0 ≤ pi ≤ 1
Domain:
min ≤ x ≤ max continuous
Density and Cumulative Functions:
pi+1 − pi f (x) = for xi ≤ x < xi+1 xi+1 − xi
⎛ x − xi ⎞ F(x) = pi + ()pi+1 − pi ⎜ ⎟ for xi ≤ x ≤ xi+1 ⎝ xi+1 − xi ⎠
With the assumptions: 1. The arrays are ordered from left to right
2. The i index runs from 0 to N+1, with two extra elements : x0 ≡ min, p0 ≡ 0 and xN+1 ≡ max, pN+1 ≡ 1.
Mean:
No Closed Form
Variance:
No Closed Form
Skewness:
No Closed Form
Kurtosis:
No Closed Form
Mode:
No Closed Form
PDF - Cumul(0,5,{1,2,3,4},{.2,.3,.7,.8}) CDF - Cumul(0,5,{1,2,3,4},{.2,.3,.7,.8})
0.45 1.0
0.40
0.8 0.35
0.30 0.6 0.25
0.20 0.4 0.15
0.10 0.2
0.05
0.00 0.0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 -1 -1
Cumulative (Descending) RISKCumulD(min, max, {x}, {p})
Parameters:
min continuous parameter min < max
max continuous parameter
{x} = {x1, x2, …, xN} array of continuous parameters min ≤ xi ≤ max
{p} = {p1, p2, …, pN} array of continuous parameters 0 ≤ pi ≤ 1
Domain:
min ≤ x ≤ max continuous
Density and Cumulative Functions:
pi − pi+1 f (x) = for xi ≤ x < xi+1 xi+1 − xi
⎛ x − xi ⎞ F(x) = 1− pi + ()pi − pi+1 ⎜ ⎟ for xi ≤ x ≤ xi+1 ⎝ xi+1 − xi ⎠
With the assumptions: 1. The arrays are ordered from left to right
2. The i index runs from 0 to N+1, with two extra elements : x0 ≡ min, p0 ≡ 1 and xN+1 ≡ max, pN+1 ≡ 0.
Mean:
No Closed Form
Variance:
No Closed Form
Skewness:
No Closed Form
Kurtosis:
No Closed Form
Mode:
No Closed Form
PDF - CumulD(0,5,{1,2,3,4},{.8,.7,.3,.2}) CDF - CumulD(0,5,{1,2,3,4},{.8,.7,.3,.2})
0.45 1.0
0.40
0.8 0.35
0.30 0.6 0.25
0.20 0.4 0.15
0.10 0.2
0.05
0.00 0.0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 -1 -1
Discrete RISKDiscrete({x}, {p})
Parameters:
{x} = {x1, x2, …, xN} array of continuous parameters
{p} = {p1, p2, …, pN} array of continuous parameters
Domain:
x ∈{x} discrete
Mass and Cumulative Functions:
f (x) = pi for x = xi
f (x) = 0 for x ∉{x}
F(x) = 0 for x < x1
s F(x) = ∑pi for xs ≤ x < xs+1, s < N i=1
F(x) = 1 for x ≥ xN
With the assumptions: 1. The arrays are ordered from left to right 2. The p array is normalized to 1.
Mean:
N ∑ xipi ≡ µ i=1
Variance:
N 2 ∑(xi − µ) pi ≡ V i=1
Skewness:
N 1 (x − µ)3 p 3 2 ∑ i i V i=1
Kurtosis:
N 1 (x − µ)4 p 2 ∑ i i V i=1
Mode:
The x-value corresponding to the highest p-value.
PMF - Discrete({1,2,3,4},{2,1,2,1}) CDF - Discrete({1,2,3,4},{2,1,2,1})
0.35 1.0
0.30 0.8
0.25
0.6 0.20
0.15 0.4
0.10
0.2 0.05
0.00 0.0 5 0 5 0 5 0 5 0 5 5 0 5 0 5 0 5 0 5 0. 1. 1. 2. 2. 3. 3. 4. 4. 0. 1. 1. 2. 2. 3. 3. 4. 4.
Discrete Uniform RISKDUniform({x})
Parameters:
{x} = {x1, x1, …, xN} array of continuous parameters
Domain:
x ∈{x} discrete
Mass and Cumulative Functions:
1 f (x) = for x ∈{x} N
f (x) = 0 for x ∉{x}
F(x) = 0 for x < x1
i F(x) = for xi ≤ x < xi+1 N
F(x) = 1 for x ≥ xN assuming the {x} array is ordered.
Mean:
N 1 x ≡ µ N ∑ i i=1
Variance:
N 1 (x − µ)2 ≡ V N ∑ i i=1
Skewness:
N 1 (x − µ)3 3 2 ∑ i NV i=1
Kurtosis:
N 1 (x − µ)4 2 ∑ i NV i=1
Mode:
Not uniquely defined
PMF - DUniform({1,5,8,11,12}) CDF - DUniform({1,5,8,11,12})
0.25 1.0
0.20 0.8
0.15 0.6
0.10 0.4
0.05 0.2
0.00 0.0 0 2 4 6 8 0 2 4 6 8 10 12 14 10 12 14
“Error Function” RISKErf(h)
Parameters:
h continuous inverse scale parameter h > 0
Domain:
-∞ < x < +∞ continuous
Density and Cumulative Functions:
h 2 f (x) = e−(hx) π
1+ erf (hx) F(x) ≡ Φ()2hx = 2 where Φ is called the Laplace-Gauss Integral and erf is the Error Function.
Mean:
0
Variance:
1 2h2
Skewness:
0
Kurtosis:
3
Mode:
0
PDF - Erf(1) CDF - Erf(1)
0.6 1.0
0.9 0.5 0.8
0.7 0.4 0.6
0.3 0.5
0.4 0.2 0.3
0.2 0.1 0.1
0.0 0.0 0 5 0 5 0 5 0 5 0 5 0 5 0 0 5 0 5 0 ...... 0. 0. 1. 1. 2. 0. 0. 1. 1. 2. -2 -1 -1 -0 -2 -1 -1 -0
Erlang RISKErlang(m,β)
Parameters:
m integral shape parameter m > 0
β continuous scale parameter β > 0
Domain:
0 ≤ x < +∞ continuous
Density and Cumulative Functions:
m−1 1 ⎛ x ⎞ −x β f (x) = ⎜ ⎟ e β (m −1)!⎝ β ⎠
m− 1 i Γx β ()m ()x β F(x) = = 1− e−x β Γ m ∑ i! () i=0 where Γ is the Gamma Function and Γx is the Incomplete Gamma Function.
Mean:
mβ
Variance:
mβ2
Skewness:
2
m
Kurtosis:
6 3 + m
Mode:
β()m −1
PDF - Erlang(2,1) CDF - Erlang(2,1)
0.40 1.0
0.9 0.35 0.8 0.30 0.7 0.25 0.6
0.20 0.5
0.4 0.15 0.3 0.10 0.2 0.05 0.1
0.00 0.0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 -1 -1
Exponential RISKExpon(β)
Parameters:
β continuous scale parameter β > 0
Domain:
0 ≤ x < +∞ continuous
Density and Cumulative Functions:
e−x β f (x) = β
F(x) = 1− e−x β
Mean:
β
Variance:
β2
Skewness:
2
Kurtosis:
9
Mode:
0
PDF - Expon(1) CDF - Expon(1)
1.2 1.0
0.9 1.0 0.8
0.7 0.8 0.6
0.6 0.5
0.4 0.4 0.3
0.2 0.2 0.1
0.0 0.0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 . . 0. 0. 1. 1. 2. 2. 3. 3. 4. 4. 5. 0. 0. 1. 1. 2. 2. 3. 3. 4. 4. 5. -0 -0
Extreme Value RISKExtValue(a, b)
Parameters:
a continuous location parameter
b continuous scale parameter b > 0
Domain:
-∞ < x < +∞ continuous
Density and Cumulative Functions:
1 ⎛ 1 ⎞ f (x) = ⎜ ⎟ b ⎝ ez+exp(−z) ⎠
1 ()x − a F(x) = where z ≡ eexp(−z) b
Mean:
a − bΓ′()1 ≈ a +.577b where Γ’(x) is the derivative of the Gamma Function.
Variance:
π2b2
6
Skewness:
12 6 ζ()3 ≈ 1.139547 π3
Kurtosis:
5.4
Mode:
a
PDF - ExtValue(0,1) CDF - ExtValue(0,1)
0.40 1.0
0.9 0.35 0.8 0.30 0.7 0.25 0.6
0.20 0.5
0.4 0.15 0.3 0.10 0.2 0.05 0.1
0.00 0.0 0 1 2 3 4 5 0 1 2 3 4 5 -2 -1 -2 -1
Gamma RISKGamma(α, β)
Parameters:
α continuous shape parameter α > 0
β continuous scale parameter β > 0
Domain:
0 < x < +∞ continuous
Density and Cumulative Functions:
α−1 1 ⎛ x ⎞ −x β f (x) = ⎜ ⎟ e βΓ()α ⎝ β ⎠
Γx β ()α F(x) = Γ()α where Γ is the Gamma Function and Γx is the Incomplete Gamma Function.
Mean:
βα
Variance:
β2α
Skewness:
2
α
Kurtosis:
6 3 + α
Mode:
β(α −1) if α ≥ 1
0 if α < 1
PDF - Gamma(4,1) CDF - Gamma(4,1)
0.25 1.0
0.9
0.20 0.8
0.7
0.15 0.6
0.5
0.10 0.4
0.3
0.05 0.2
0.1
0.00 0.0 0 2 4 6 8 0 2 4 6 8 -2 -2 10 12 10 12
General RISKGeneral(min, max, {x}, {p})
Parameters:
min continuous parameter min < max
max continuous parameter
{x} = {x1, x2, …, xN} array of continuous parameters min ≤ xi ≤ max
{p} = {p1, p2, …, pN} array of continuous parameters pi ≥ 0
Domain:
min ≤ x ≤ max continuous
Density and Cumulative Functions:
⎡ x − xi ⎤ f (x) = pi + ⎢ ⎥(pi+1 − pi ) for xi ≤ x ≤ xi+1 ⎣xi+1 − xi ⎦
⎡ (pi+1 − pi )(x − xi )⎤ F(x) = F(xi ) + ()x − xi ⎢pi + ⎥ for xi ≤ x ≤ xi+1 ⎣ 2(xi+1 − xi )⎦
With the assumptions: 1. The arrays are ordered from left to right 2. The {p} array has been normalized to give the general distribution unit area.
3. The i index runs from 0 to N+1, with two extra elements : x0 ≡ min, p0 ≡ 0 and xN+1 ≡ max, pN+1 ≡ 0.
Mean:
No Closed Form
Variance:
No Closed Form
Skewness:
No Closed Form
Kurtosis:
No Closed Form
Mode:
No Closed Form
PDF - General(0,5,{1,2,3,4},{2,1,2,1}) CDF - General(0,5,{1,2,3,4},{2,1,2,1})
0.35 1.0
0.30 0.8 0.25
0.6 0.20
0.15 0.4
0.10
0.2 0.05
0.00 0.0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 -1 -1
Geometric RISKGeomet(p)
Parameters:
p continuous “success” probability 0< p ≤ 1
Domain:
0 ≤ x < +∞ discrete integers
Mass and Cumulative Functions:
x f (x) = p(1− p)
F(x) = 1− (1− p)x +1
Mean:
1 −1 p
Variance:
1− p
p2
Skewness:
()2 − p for p < 1 1− p
Not Defined for p = 1
Kurtosis: p2 9 + for p < 1 1− p
Not Defined for p = 1
Mode:
0
PMF - Geomet(.5) CDF - Geomet(.5)
0.6 1.0
0.9 0.5 0.8
0.7 0.4 0.6
0.3 0.5
0.4 0.2 0.3
0.2 0.1 0.1
0.0 0.0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 -1 -1
Histogram RISKHistogrm(min, max, {p})
Parameters:
min continuous parameter min < max *
max continuous parameter
{p} = {p1, p2, …, pN} array of continuous parameters pi ≥ 0
* min = max is supported for modelling convenience, but yields a degenerate distribution.
Domain:
min ≤ x ≤ max continuous
Density and Cumulative Functions:
f (x) = pi for xi ≤ x < xi+1
⎛ x − xi ⎞ F(x) = F(xi ) + pi ⎜ ⎟ for xi ≤ x ≤ xi+1 ⎝ xi+1 − xi ⎠
⎛ max− min ⎞ where xi ≡ min+ i⎜ ⎟ ⎝ N ⎠
The {p} array has been normalized to give the histogram unit area.
Mean:
No Closed Form
Variance:
No Closed Form
Skewness:
No Closed Form
Kurtosis:
No Closed Form
Mode:
Not Uniquely Defined.
PDF - Histogrm(0,5,{6,5,3,4,5}) CDF - Histogrm(0,5,{6,5,3,4,5})
0.30 1.0
0.25 0.8
0.20 0.6
0.15
0.4 0.10
0.2 0.05
0.00 0.0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 -1 -1
Hypergeometric RISKHyperGeo(n, D, M)
Parameters:
n the number of draws integer 0 ≤ n ≤ M
D the number of "tagged" items integer 0 ≤ D ≤ M
M the total number of items integer M ≥ 0
Domain:
max(0,n+D-M) ≤ x ≤ min(n,D) discrete integers
Mass and Cumulative Functions:
⎛D⎞⎛M − D⎞ ⎛D⎞⎛M − D⎞ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ x ⎟⎜ n − x ⎟ x ⎜ x ⎟⎜ n − x ⎟ f (x) = ⎝ ⎠⎝ ⎠ F(x) = ⎝ ⎠⎝ ⎠ ⎛M⎞ ∑ ⎛M⎞ ⎜ ⎟ i=1 ⎜ ⎟ ⎝ n ⎠ ⎝ n ⎠
Mean:
nD for M > 0 M
0 for M = 0
Variance:
nD ⎡()M − D (M − n)⎤ ⎢ ⎥ for M>1 M2 ⎣ ()M −1 ⎦
0 for M = 1
Skewness:
()M − 2D (M − 2n) M −1 for M>2, M>D>0, M>n>0 M − 2 nD()M − D (M − n)
Not Defined otherwise
Kurtosis:
M2 ()M −1 ⎡M(M +1)− 6n(M − n) 3n(M − n)(M + 6) ⎤ ⎢ + − 6⎥ n()M − 2 (M − 3)(M − n)⎣ D()M − D M2 ⎦
for M>3, M>D>0, M>n>0
Not Defined otherwise
Mode:
(bimodal) xm and xm-1 if xm is integral
(unimodal) biggest integer less than xm otherwise
()n +1 (D +1) where x ≡ m M + 2
PMF - HyperGeo(6,5,10) CDF - HyperGeo(6,5,10)
0.50 1.0
0.45
0.40 0.8
0.35
0.30 0.6
0.25
0.20 0.4
0.15
0.10 0.2
0.05
0.00 0.0 0 1 2 3 4 5 6 0 1 2 3 4 5 6
Integer Uniform RISKIntUniform(min, max)
Parameters:
min discrete boundary parameter min < max
max discrete boundary probability
Domain:
min ≤ x ≤ max discrete integers
Mass and Cumulative Functions:
1 f (x) = max− min+1
x − min+1 F(x) = max− min+1
Mean:
min+ max
2
Variance:
∆()∆ + 2 where ∆≡(max-min) 12
Skewness:
0
Kurtosis:
⎛ 9 ⎞ ⎛ n 2 − 7 / 3⎞ ⎜ ⎟ ⋅⎜ ⎟ where n≡(max-min+1) ⎜ 2 ⎟ ⎝ 5 ⎠ ⎝ n −1 ⎠
Mode:
Not uniquely defined
PMF - IntUniform(0,8) CDF - IntUniform(0,8)
0.12 1.0
0.10 0.8
0.08 0.6
0.06
0.4 0.04
0.2 0.02
0.00 0.0 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 -1 -1
Inverse Gaussian RISKInvGauss(µ, λ)
Parameters:
µ continuous parameter µ > 0
λ continuous parameter λ > 0
Domain:
x > 0 continuous
Density and Cumulative Functions:
⎡λ()x −µ 2 ⎤ −⎢ ⎥ λ ⎢ 2 ⎥ f (x) = e ⎣ 2µ x ⎦ 2π x3
⎡ λ ⎛ x ⎞⎤ 2λ µ ⎡ λ ⎛ x ⎞⎤ F(x) = Φ⎢ ⎜ −1⎟⎥ + e Φ⎢− ⎜ +1⎟⎥ ⎣ x ⎝ µ ⎠⎦ ⎣ x ⎝ µ ⎠⎦ where Φ(z) is the cumulative distribution function of a Normal(0,1), also called the Laplace-Gauss Integral
Mean:
µ
Variance:
µ3
λ
Skewness:
µ 3 λ
Kurtosis:
µ 3 +15 λ
Mode:
⎡ 9µ2 3µ ⎤ µ ⎢ 1+ − ⎥ ⎢ 2 2λ⎥ ⎣ 4λ ⎦
PDF - InvGauss(1,2) CDF - InvGauss(1,2)
1.2 1.0
0.9 1.0 0.8
0.7 0.8 0.6
0.6 0.5
0.4 0.4 0.3
0.2 0.2 0.1
0.0 0.0 5 5 0 5 0 5 0 5 0 5 0 0 5 0 5 0 5 0 5 0 . . 0. 0. 1. 1. 2. 2. 3. 3. 4. 0. 0. 1. 1. 2. 2. 3. 3. 4. -0 -0
Logistic RISKLogistic(α, β)
Parameters:
α continuous location parameter
β continuous scale parameter β > 0
Domain:
-∞ < x < +∞ continuous
Density and Cumulative Functions:
⎛ 1 ⎛ x − α ⎞⎞ 2 ⎜ ⎟ sec h ⎜ ⎜ ⎟⎟ 2 ⎝ β ⎠ f (x) = ⎝ ⎠ 4β
⎛ 1 ⎛ x − α ⎞⎞ ⎜ ⎟ 1+ tanh⎜ ⎜ ⎟⎟ 2 ⎝ β ⎠ F(x) = ⎝ ⎠ 2 where “sech” is the Hyperbolic Secant Function and “tanh” is the Hyperbolic Tangent Function.
Mean:
α
Variance:
π2β2
3
Skewness:
0
Kurtosis:
4.2
Mode:
α
PDF - Logistic(0,1) CDF - Logistic(0,1)
0.30 1.0
0.9 0.25 0.8
0.7 0.20 0.6
0.15 0.5
0.4 0.10 0.3
0.2 0.05 0.1
0.00 0.0 0 1 2 3 4 5 0 1 2 3 4 5 -5 -4 -3 -2 -1 -5 -4 -3 -2 -1
Log-Logistic RISKLogLogistic(γ, β, α)
Parameters:
γ continuous location parameter
β continuous scale parameter β > 0
α continuous shape parameter α > 0
Definitions:
π θ ≡ α
Domain:
γ ≤ x < +∞ continuous
Density and Cumulative Functions:
α tα−1 f (x) = 2 β(1+ tα )
1 x − γ F(x) = with t ≡ α ⎛1⎞ β 1+ ⎜ ⎟ ⎝ t ⎠
Mean:
βθ csc()θ + γ for α > 1
Variance:
β2θ [2csc()2θ − θcsc2 (θ)] for α > 2
Skewness:
3csc()3θ − 6θcsc(2θ)csc(θ)+ 2θ2 csc3(θ) for α > 3 3 θ[]2csc()2θ − θcsc2 (θ)2
Kurtosis:
4csc()4θ −12θcsc(3θ)csc(θ)+12θ2 csc(2θ)csc2 (θ)− 3θ3 csc4 (θ) for α > 4 2 θ []2csc()2θ − θcsc2 (θ)
Mode: 1 ⎡α −1⎤ α γ + β ⎢ ⎥ for α > 1 ⎣α +1⎦
γ for α ≤ 1
PDF - LogLogistic(0,1,5) CDF - LogLogistic(0,1,5)
1.4 1.0
0.9 1.2 0.8
1.0 0.7
0.6 0.8 0.5 0.6 0.4
0.4 0.3
0.2 0.2 0.1
0.0 0.0 5 5 0 5 0 5 0 5 0 0 5 0 5 0 5 0 . . 0. 0. 1. 1. 2. 2. 3. 0. 0. 1. 1. 2. 2. 3. -0 -0
Lognormal (Format 1) RISKLognorm(µ, σ)
Parameters:
µ continuous parameter µ > 0
σ continuous parameter σ > 0
Domain:
0 ≤ x < +∞ continuous
Density and Cumulative Functions:
2 1 ⎡ln x −µ′⎤ 1 − ⎢ ⎥ f (x) = e 2 ⎣ σ′ ⎦ x 2πσ′
⎛ ln x − µ′ ⎞ F(x) = Φ⎜ ⎟ ⎝ σ′ ⎠
⎡ 2 ⎤ ⎡ 2 ⎤ ⎢ µ ⎥ ⎛ σ ⎞ with µ′ ≡ ln and σ′ ≡ ln⎢1+ ⎜ ⎟ ⎥ ⎢ 2 2 ⎥ ⎢ µ ⎥ ⎣ σ + µ ⎦ ⎣ ⎝ ⎠ ⎦ where Φ(z) is the cumulative distribution function of a Normal(0,1) also called the Laplace-Gauss Integral.
Mean:
µ
Variance:
σ2
Skewness:
3 ⎛ σ ⎞ ⎛ σ ⎞ ⎜ ⎟ + 3⎜ ⎟ ⎝ µ ⎠ ⎝ µ ⎠
Kurtosis:
2 4 3 2 ⎛ σ ⎞ ω + 2ω + 3ω − 3 with ω ≡ 1+ ⎜ ⎟ ⎝ µ ⎠
Mode:
µ4
3 2 (σ2 + µ2 )
PDF - Lognorm(1,1) CDF - Lognorm(1,1)
1.0 1.0
0.9 0.9
0.8 0.8
0.7 0.7
0.6 0.6
0.5 0.5
0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0.0 0.0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 -1 -1
Lognormal (Format 2) RISKLognorm2(µ, σ)
Parameters:
µ continuous parameter
σ continuous parameter σ > 0
Domain:
0 ≤ x < +∞ continuous
Density and Cumulative Functions:
2 1 ⎡ln x−µ ⎤ 1 − ⎢ ⎥ f (x) = e 2 ⎣ σ ⎦ x 2πσ
⎛ ln x − µ ⎞ F(x) = Φ⎜ ⎟ ⎝ σ ⎠ where Φ(z) is the cumulative distribution function of a Normal(0,1), also called the Laplace-Gauss Integral
Mean:
σ2 µ+ e 2
Variance:
2 e2µω(ω −1) with ω ≡ eσ
Skewness:
2 ()ω + 2 ω −1 with ω ≡ eσ
Kurtosis:
2 ω4 + 2ω3 + 3ω2 − 3 with ω ≡ eσ
Mode:
2 eµ−σ
PDF - Lognorm2(0,1) CDF - Lognorm2(0,1)
0.7 1.0
0.9 0.6 0.8
0.5 0.7
0.6 0.4 0.5 0.3 0.4
0.2 0.3
0.2 0.1 0.1
0.0 0.0 0 2 4 6 8 0 2 4 6 8 -2 -2 10 12 10 12
Negative Binomial RISKNegBin(s, p)
Parameters:
s the number of successes discrete parameter s ≥ 0
p probability of a single success continuous parameter 0 < p ≤ 1
Domain:
0 ≤ x < +∞ discrete integers
Density and Cumulative Functions:
⎛s + x −1⎞ s x f (x) = ⎜ ⎟p (1− p) ⎝ x ⎠
x ⎛s + i −1⎞ F(x) = ps ⎜ ⎟(1− p)i ∑⎜ i ⎟ i=0 ⎝ ⎠
Where ( ) is the Binomial Coefficient.
Mean:
s()1− p
p
Variance:
s()1− p
p2
Skewness:
2 − p for s > 0, p < 1 s()1− p
Kurtosis:
6 p2 3 + + for s > 0, p < 1 s s()1− p
Mode:
(bimodal) z and z + 1 integer z > 0
(unimodal) 0 z < 0
(unimodal) smallest integer greater than z otherwise
s()1− p −1 where z ≡ p
PDF - NegBin(3,.6) CDF - NegBin(3,.6)
0.30 1.0
0.9 0.25 0.8
0.7 0.20 0.6
0.15 0.5
0.4 0.10 0.3
0.2 0.05 0.1
0.00 0.0 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 -1 -1
Normal RISKNormal(µ, σ)
Parameters:
µ continuous location parameter
σ continuous scale parameter σ > 0 *
*σ = 0 is supported for modeling convenience, but gives a degenerate distribution with x = µ.
Domain:
-∞ < x < +∞ continuous
Density and Cumulative Functions:
2 1⎛ x−µ ⎞ 1 − ⎜ ⎟ f (x) = e 2⎝ σ ⎠ 2πσ
⎛ x − µ ⎞ 1 ⎡ ⎛ x − µ ⎞ ⎤ F(x) ≡ Φ⎜ ⎟ = ⎢erf⎜ ⎟ +1⎥ ⎝ σ ⎠ 2 ⎣ ⎝ 2σ ⎠ ⎦ where Φ is called the Laplace-Gauss Integral and erf is the Error Function.
Mean:
µ
Variance:
σ2
Skewness:
0
Kurtosis:
3
Mode:
µ
PDF - Normal(0,1) CDF - Normal(0,1)
0.45 1.0
0.40 0.9
0.8 0.35 0.7 0.30 0.6 0.25 0.5 0.20 0.4 0.15 0.3 0.10 0.2
0.05 0.1
0.00 0.0 0 1 2 3 0 1 2 3 -3 -2 -1 -3 -2 -1
Pareto (First Kind) Pareto(θ, a)
Parameters:
θ continuous shape parameter θ > 0
a continuous scale parameter a > 0
Domain:
a ≤ x < +∞ continuous
Density and Cumulative Functions:
θa θ f (x) = xθ+1
θ ⎛ a ⎞ F(x) = 1− ⎜ ⎟ ⎝ x ⎠
Mean:
aθ for θ > 1 θ −1
Variance:
θa 2 for θ > 2 ()θ −1 2 (θ − 2)
Skewness:
θ +1 θ − 2 2 for θ > 3 θ − 3 θ
Kurtosis:
3()θ − 2 3θ2 + θ + 2 ( ) for θ > 4 θ()θ − 3 (θ − 4)
Mode:
a
PDF - Pareto(2,1) CDF - Pareto(2,1)
2.0 1.0
1.8 0.9
1.6 0.8
1.4 0.7
1.2 0.6
1.0 0.5
0.8 0.4
0.6 0.3
0.4 0.2
0.2 0.1
0.0 0.0 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 10 11 10 11
Pareto (Second Kind) RISKPareto2(b, q)
Parameters:
b continuous scale parameter b > 0
q continuous shape parameter q > 0
Domain:
0 ≤ x < +∞ continuous
Density and Cumulative Functions:
qbq f (x) = ()x + b q+1
bq F(x) = 1− ()x + b q
Mean:
b for q > 1 q −1
Variance:
b2q for q > 2 ()q −1 2 (q − 2)
Skewness:
⎡ q +1⎤ q − 2 2 ⎢ ⎥ for q > 3 ⎣q − 3⎦ q
Kurtosis:
3()q − 2 3q2 + q + 2 ( ) for q > 4 q()q − 3 (q − 4)
Mode:
0
PDF - Pareto2(3,3) CDF - Pareto2(3,3)
1.2 1.0
0.9 1.0 0.8
0.7 0.8 0.6
0.6 0.5
0.4 0.4 0.3
0.2 0.2 0.1
0.0 0.0 0 2 4 6 8 0 2 4 6 8 -2 -2 10 12 10 12
Pearson Type V RISKPearson5(α, β)
Parameters:
α continuous shape parameter α > 0
β continuous scale parameter β > 0
Domain:
0 ≤ x < +∞ continuous
Density and Cumulative Functions:
1 e−β x f (x) = ⋅ βΓ()α ()x β α+1
F(x) Has No Closed Form
Mean:
β for α > 1 α −1
Variance:
β2 for α > 2 ()α −1 2 (α − 2)
Skewness:
4 α − 2 for α > 3 α − 3
Kurtosis:
3()α + 5 (α − 2) for α > 4 ()α − 3 (α − 4)
Mode:
β
α +1
PDF - Pearson5(3,1) CDF - Pearson5(3,1)
2.5 1.0
0.9
2.0 0.8
0.7
1.5 0.6
0.5
1.0 0.4
0.3
0.5 0.2
0.1
0.0 0.0 5 5 0 5 0 5 0 5 0 5 0 5 0 5 . . 0. 0. 1. 1. 2. 2. 0. 0. 1. 1. 2. 2. -0 -0
Pearson Type VI
RISKPearson6(α1, α2, β)
Parameters:
α1 continuous shape parameter α1 > 0
α2 continuous shape parameter α2 > 0
β continuous scale parameter β > 0
Domain:
0 ≤ x < +∞ continuous
Density and Cumulative Functions:
1 ()x β α1−1 f (x) = × α1+α2 βB()α1,α 2 ⎛ x ⎞ ⎜1+ ⎟ ⎝ β ⎠
F(x) Has No Closed Form. where B is the Beta Function.
Mean:
βα1 for α2 > 1 α 2 −1
Variance:
β2α ()α + α −1 1 1 2 for α > 2 2 2 ()α2 −1 (α2 − 2)
Skewness:
α 2 − 2 ⎡2α1 + α 2 −1⎤ 2 ⎢ ⎥ for α2 > 3 α1()α1 + α 2 −1 ⎣ α 2 − 3 ⎦
Kurtosis:
⎡ 2 ⎤ 3()α 2 − 2 2 ()α 2 −1 ⎢ + (α 2 + 5)⎥ for α2 > 4 ()α 2 − 3 (α 2 − 4)⎣⎢α1()α1 + α 2 −1 ⎦⎥
Mode:
β()α1 −1 for α1 > 1 α 2 +1
0 otherwise
PDF - Pearson6(3,3,1) CDF - Pearson6(3,3,1)
0.7 1.0
0.9 0.6 0.8
0.5 0.7
0.6 0.4 0.5 0.3 0.4
0.2 0.3
0.2 0.1 0.1
0.0 0.0 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 -1 -1
Pert (Beta) RISKPert(min, m.likely, max)
Definitions:
min+ 4 ⋅ m.likely + max ⎡ µ − min ⎤ ⎡ max− µ ⎤ µ ≡ α1 ≡ 6 α2 ≡ 6 6 ⎣⎢max− min ⎦⎥ ⎣⎢max− min ⎦⎥
Parameters:
min continuous boundary parameter min < max
m.likely continuous parameter min < m.likely < max
max continuous boundary parameter
Domain:
min ≤ x ≤ max continuous
Density and Cumulative Functions:
()x − min α1 −1(max− x)α2 −1 f (x) = α1 +α2 −1 Β(α1,α2 )()max− min
Bz ()α1,α2 x − min F(x) = ≡ Iz (α1,α2 ) with z ≡ B(α1,α2 ) max− min where B is the Beta Function and Bz is the Incomplete Beta Function.
Mean:
min+ 4 ⋅ m.likely + max µ ≡ 6
Variance:
()µ − min (max− µ)
7
Skewness:
min+ max− 2µ 7
4 ()µ − min (max− µ)
Kurtosis:
()α + α +1 2(α + α )2 + α α (α + α − 6) 3 1 2 ( 1 2 1 2 1 2 ) α1α2 ()α1 + α2 + 2 (α1 + α2 + 3)
Mode:
m.likely
PDF - Pert(0,1,3) CDF - Pert(0,1,3)
0.7 1.0
0.6 0.8
0.5
0.6 0.4
0.3 0.4
0.2
0.2 0.1
0.0 0.0 5 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 . . 0. 0. 1. 1. 2. 2. 3. 3. 0. 0. 1. 1. 2. 2. 3. 3. -0 -0
Poisson RISKPoisson(λ)
Parameters:
λ mean number of successes continuous λ > 0 *
*λ = 0 is supported for modeling convenience, but gives a degenerate distribution with x = 0.
Domain:
0 ≤ x < +∞ discrete integers
Mass and Cumulative Functions:
λxe−λ f (x) = x!
x λn F(x) = e−λ ∑ n! n =0
Mean:
λ
Variance:
λ
Skewness:
1
λ
Kurtosis:
1 3 + λ
Mode:
(bimodal) λ and λ-1 (bimodal) if λ is an integer
(unimodal) largest integer less than λ otherwise
PMF - Poisson(3) CDF - Poisson(3)
0.25 1.0
0.9
0.20 0.8
0.7
0.15 0.6
0.5
0.10 0.4
0.3
0.05 0.2
0.1
0.00 0.0 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 -1 -1
Rayleigh RISKRayleigh(b)
Parameters:
b continuous scale parameter b > 0
Domain:
0 ≤ x < +∞ continuous
Density and Cumulative Functions:
2 1 ⎛ x ⎞ x − ⎜ ⎟ f (x) = e 2 ⎝ b ⎠ b2
2 1 ⎛ x ⎞ − ⎜ ⎟ F(x) = 1− e 2 ⎝ b ⎠
Mean:
π b 2
Variance:
⎛ π ⎞ b2 ⎜2 − ⎟ ⎝ 2 ⎠
Skewness:
2()π − 3 π ≈ 0.6311 ()4 − π 3 2
Kurtosis:
32 − 3π2 ≈ 3.2451 ()4 − π 2
Mode:
b
PDF - Rayleigh(1) CDF - Rayleigh(1)
0.7 1.0
0.9 0.6 0.8
0.5 0.7
0.6 0.4 0.5 0.3 0.4
0.2 0.3
0.2 0.1 0.1
0.0 0.0 5 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 . . 0. 0. 1. 1. 2. 2. 3. 3. 0. 0. 1. 1. 2. 2. 3. 3. -0 -0
Student’s “t” RISKStudent(ν)
Parameters:
ν the degrees of freedom integer ν > 0
Domain:
-∞ < x < +∞ continuous
Density and Cumulative Functions:
⎛ ν +1⎞ Γ⎜ ⎟ ν+1 1 ⎝ 2 ⎠ ⎡ ν ⎤ 2 f (x) = ⎢ ⎥ πν ⎛ ν ⎞ 2 Γ⎜ ⎟ ⎣ν + x ⎦ ⎝ 2 ⎠
1 ⎡ ⎛ 1 ν ⎞⎤ x2 F(x) = ⎢1+ Is ⎜ , ⎟⎥ with s ≡ 2 ⎣ ⎝ 2 2 ⎠⎦ ν + x2 where Γ is the Gamma Function and Ix is the Incomplete Beta Function.
Mean:
0 for ν > 1*
*even though the mean is not defined for ν = 1, the distribution is still symmetrical about 0.
Variance:
ν for ν > 2 ν − 2
Skewness:
0 for ν > 3*
*even though the skewness is not defined for ν ≤ 3, the distribution is still symmetric about 0.
Kurtosis:
⎛ ν − 2 ⎞ 3⎜ ⎟ for ν > 4 ⎝ ν − 4 ⎠
Mode:
0
PDF - Student(3) CDF - Student(3)
0.40 1.0
0.9 0.35 0.8 0.30 0.7 0.25 0.6
0.20 0.5
0.4 0.15 0.3 0.10 0.2 0.05 0.1
0.00 0.0 0 1 2 3 4 5 0 1 2 3 4 5 -5 -4 -3 -2 -1 -5 -4 -3 -2 -1
Triangular RISKTriang(min, m.likely, max)
Parameters:
min continuous boundary parameter min < max *
m.likely continuous mode parameter min ≤ m.likely ≤ max
max continuous boundary parameter
*min = max is supported for modeling convenience, but gives a degenerate distribution.
Domain:
min ≤ x ≤ max continuous
Density and Cumulative Functions:
2()x − min f (x) = min ≤ x ≤ m.likely (m.likely − min)(max− min)
2()max− x f (x) = m.likely ≤ x ≤ max (max− m.likely)(max− min)
()x − min 2 F(x) = min ≤ x ≤ m.likely (m.likely − min)(max− min)
()max− x 2 F(x) = 1− m.likely ≤ x ≤ max (max− m.likely)(max− min)
Mean:
min+ m.likely + max
3
Variance:
min2 + m.likely2 + max2 − (max)(m.likely) − (m.likely)(min) − (max)(min)
18
Skewness:
2 2 f f 2 − 9 2(m.likely − min) ( ) where f ≡ −1 5 3 2 max− min (f 2 + 3)
Kurtosis:
2.4
Mode:
m.likely
PDF - Triang(0,3,5) CDF - Triang(0,3,5)
0.45 1.0
0.40
0.8 0.35
0.30 0.6 0.25
0.20 0.4 0.15
0.10 0.2
0.05
0.00 0.0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 -1 -1
Uniform RISKUniform(min, max)
Parameters:
min continuous boundary parameter min < max *
max continuous boundary parameter
*min = max is supported for modeling convenience, but gives a degenerate distribution.
Domain:
min ≤ x ≤ max continuous
Density and Cumulative Functions:
1 f (x) = max− min
x − min F(x) = max− min
Mean:
max− min
2
Variance:
()max− min 2
12
Skewness:
0
Kurtosis:
1.8
Mode:
Not uniquely defined
PDF - Uniform(0,1) CDF - Uniform(0,1)
1.2 1.0
1.0 0.8
0.8 0.6
0.6
0.4 0.4
0.2 0.2
0.0 0.0 2 2 0 2 4 6 8 0 2 0 2 4 6 8 0 2 . . 0. 0. 0. 0. 0. 1. 1. 0. 0. 0. 0. 0. 1. 1. -0 -0
Weibull RISKWeibull(α, β)
Parameters:
α continuous shape parameter α > 0
β continuous scale parameter β > 0
Domain:
0 ≤ x < +∞ continuous
Density and Cumulative Functions:
αxα−1 α f (x) = e−(x β) βα
α F(x) = 1− e−(x β)
Mean:
⎛ 1 ⎞ βΓ⎜1+ ⎟ ⎝ α ⎠ where Γ is the Gamma Function.
Variance:
2 ⎡ ⎛ 2 ⎞ 2 ⎛ 1 ⎞⎤ β ⎢Γ⎜1+ ⎟ − Γ ⎜1+ ⎟⎥ ⎣ ⎝ α ⎠ ⎝ α ⎠⎦ where Γ is the Gamma Function.
Skewness:
⎛ 3 ⎞ ⎛ 2 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ Γ⎜1+ ⎟ − 3Γ⎜1+ ⎟Γ⎜1+ ⎟ + 2Γ3⎜1+ ⎟ α α α α ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 3 2 ⎡ ⎛ 2 ⎞ 2 ⎛ 1 ⎞⎤ ⎢Γ⎜1+ ⎟ − Γ ⎜1+ ⎟⎥ ⎣ ⎝ α ⎠ ⎝ α ⎠⎦ where Γ is the Gamma Function.
Kurtosis:
⎛ 4 ⎞ ⎛ 3 ⎞ ⎛ 1 ⎞ ⎛ 2 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ Γ⎜1+ ⎟ − 4Γ⎜1+ ⎟Γ⎜1+ ⎟ + 6Γ⎜1+ ⎟Γ2 ⎜1+ ⎟ − 3Γ4 ⎜1+ ⎟ α α α α α α ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 2 ⎡ ⎛ 2 ⎞ 2 ⎛ 1 ⎞⎤ ⎢Γ⎜1+ ⎟ − Γ ⎜1+ ⎟⎥ ⎣ ⎝ α ⎠ ⎝ α ⎠⎦ where Γ is the Gamma Function.
Mode: 1 α ⎛ 1 ⎞ β⎜1− ⎟ for α >1 ⎝ α ⎠
0 for α ≤ 1
PDF - Weibull(2,1) CDF - Weibull(2,1)
0.9 1.0
0.8 0.9
0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2
0.1 0.1
0.0 0.0 0 5 0 5 0 5 0 5 0 5 0 5 5 5 . . 0. 0. 1. 1. 2. 2. 0. 0. 1. 1. 2. 2. -0 -0