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DistFunc.pdf
```A Concise Summary of @RISK Probability
Distribution Functions
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:
f (x) =
x α1 −1 (1 − x )α 2 −1
Β(α1 , α 2 )
F( x ) =
B x (α1 , α 2 )
≡ I x (α1 , α 2 )
B(α1 , α 2 )
where B is the Beta Function and Bx is the Incomplete Beta Function.
Mean:
α1
α1 + α 2
Variance:
α1α 2
(α1 + α 2 )2 (α1 + α 2 + 1)
Skewness:
2
α 2 − α1
α1 + α 2 + 1
α1 + α 2 + 2
α1α 2
Kurtosis:
3
(α1 + α 2 + 1)(2(α1 + α 2 )2 + α1α 2 (α1 + α 2 − 6))
α1α 2 (α1 + α 2 + 2)(α1 + α 2 + 3)
Mode:
α1 − 1
α1 + α 2 − 2
α1>1, α2>1
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
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.0
-0.2
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:
f (x) =
(x − min )α1 −1 (max− x )α 2 −1
Β(α1 , α 2 )(max − min )α1 + α 2 −1
F( x ) =
B z (α1 , α 2 )
≡ I z (α1 , α 2 )
B(α1 , α 2 )
where B is the Beta Function and Bz is the Incomplete Beta Function.
Mean:
min +
α1
(max− min )
α1 + α 2
Variance:
α1α 2
(α1 + α 2 ) (α1 + α 2 + 1)
2
(max− min ) 2
with z ≡
x − min
max − min
Skewness:
2
α 2 − α1
α1 + α 2 + 1
α1 + α 2 + 2
α1α 2
Kurtosis:
3
(α1 + α 2 + 1)(2(α1 + α 2 )2 + α1α 2 (α1 + α 2 − 6))
α1α 2 (α1 + α 2 + 2)(α1 + α 2 + 3)
Mode:
min +
α1 − 1
(max− min )
α1 + α 2 − 2
α1>1, α2>1
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
6
5
4
3
2
1
0
-1
6
5
4
3
2
1
0
0.0
-1
0.00
Beta (Subjective)
RISKBetaSubj(min, m.likely, mean, max)
Definitions:
mid ≡
min + max
2
α1 ≡ 2
(mean − min )(mid − m.likely)
(mean − m.likely)(max − min )
α 2 ≡ α1
max − mean
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
mean < mid
mean = mid
Domain:
min ≤ x ≤ max
if m.likely > mean
if m.likely < mean
if m.likely = mean
continuous
Density and Cumulative Distribution Functions:
f (x) =
(x − min )α1 −1 (max− x )α 2 −1
Β(α1 , α 2 )(max − min )α1 + α 2 −1
F( x ) =
B z (α1 , α 2 )
≡ I z (α1 , α 2 )
B(α1 , α 2 )
where B is the Beta Function and Bz is the Incomplete Beta Function.
with z ≡
x − min
max − min
Mean:
mean
Variance:
(mean − min )(max − mean )(mean − m.likely)
2 ⋅ mid + mean − 3 ⋅ m.likely
Skewness:
2 (mid − mean )
mean + mid − 2 ⋅ m.likely
(mean − m.likely)(2 ⋅ mid + mean − 3 ⋅ m.likely)
(mean − min )(max − mean )
Kurtosis:
(
α1 + α 2 + 1)(2(α1 + α 2 )2 + α1α 2 (α1 + α 2 − 6))
3
α1α 2 (α1 + α 2 + 2)(α1 + α 2 + 3)
Mode:
m.likely
PDF - BetaSubj(0,1,2,5)
CDF - BetaSubj(0,1,2,5)
1.0
0.30
0.25
0.8
0.20
0.6
0.15
0.4
0.10
0.2
0.05
6
5
4
3
2
1
0
-1
6
5
4
3
2
1
0
0.0
-1
0.00
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⎞
f ( x ) = ⎜⎜ ⎟⎟p x (1 − p )n − x
⎝x⎠
x
F( x ) =
⎛n⎞
∑ ⎜⎜⎝ i ⎟⎟⎠ pi (1 − p) n − i
i=0
Mean:
np
Variance:
np(1 − p )
Skewness:
(1 − 2p )
np(1 − p )
Kurtosis:
3−
6
1
+
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)
1.0
0.30
0.25
0.8
0.20
0.6
0.15
0.4
0.10
0.2
0.05
9
8
7
6
5
4
3
2
1
0
-1
9
8
7
6
5
4
3
2
1
0
0.0
-1
0.00
Chi-Squared
RISKChiSq(ν)
Parameters:
ν
discrete shape parameter
ν>0
Domain:
0 ≤ x < +∞
continuous
Density and Cumulative Functions:
f (x) =
F( x ) =
1
2
ν2
Γ(ν 2)
e − x 2 x (ν 2 )−1
Γx 2 (ν 2 )
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
16
14
12
10
8
6
4
2
0
16
14
12
10
8
6
4
0.0
2
0.00
0
0.1
-2
0.02
-2
0.2
Cumulative (Ascending)
RISKCumul(min, max, {x}, {p})
Parameters:
min
continuous parameter
max
continuous parameter
{x} = {x1, x2, …, xN}
array of continuous parameters
{p} = {p1, p2, …, pN} array of continuous parameters
min < max
min ≤ xi ≤ max
0 ≤ pi ≤ 1
Domain:
min ≤ x ≤ max
continuous
Density and Cumulative Functions:
p − pi
f ( x ) = i +1
x i +1 − x i
⎛ x − xi
F( x ) = p i + (p i +1 − p i )⎜⎜
⎝ x i +1 − x i
for xi ≤ x < xi+1
⎞
⎟⎟
⎠
for xi ≤ x ≤ xi+1
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
6
5
4
3
2
1
0
-1
6
5
4
3
2
1
0
-1
0.0
Cumulative (Descending)
RISKCumulD(min, max, {x}, {p})
Parameters:
min
continuous parameter
max
continuous parameter
{x} = {x1, x2, …, xN}
array of continuous parameters
{p} = {p1, p2, …, pN} array of continuous parameters
min < max
min ≤ xi ≤ max
0 ≤ pi ≤ 1
Domain:
min ≤ x ≤ max
continuous
Density and Cumulative Functions:
p − p i +1
f (x) = i
x i +1 − x i
⎛ x − xi
F( x ) = 1 − p i + (p i − p i +1 )⎜⎜
⎝ x i +1 − x i
for xi ≤ x < xi+1
⎞
⎟⎟
⎠
for xi ≤ x ≤ xi+1
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
6
5
4
3
2
1
0
-1
6
5
4
3
2
1
0
-1
0.0
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) = p i
for x = x i
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
With the assumptions:
1. The arrays are ordered from left to right
2. The p array is normalized to 1.
Mean:
N
∑ x i pi ≡ µ
i =1
for x ≥ xN
Variance:
N
∑ ( x i − µ) 2 p i ≡ V
i =1
Skewness:
N
1
V
32
∑ ( x i − µ) 3 p i
i =1
Kurtosis:
1
2
N
∑ ( x i − µ) 4 p 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
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.0
0.5
0.00
Discrete Uniform
RISKDUniform({x})
Parameters:
{x} = {x1, x1, …, xN}
array of continuous parameters
Domain:
x ∈ {x}
discrete
Mass and Cumulative Functions:
f (x) =
1
N
for x ∈ {x}
f (x) = 0
for x ∉ {x}
F( x ) = 0
for x < x1
F( x ) =
i
N
F( x ) = 1
assuming the {x} array is ordered.
Mean:
1
N
N
∑ xi ≡ µ
i =1
for xi ≤ x < xi+1
for x ≥ xN
Variance:
1
N
N
∑ ( x i − µ) 2 ≡ V
i =1
Skewness:
N
∑ ( x i − µ) 3
1
NV
32
i =1
Kurtosis:
N
1
2
∑ ( x i − µ) 4
NV i =1
Mode:
Not uniquely defined
14
12
10
8
0.0
6
0.00
4
0.2
0
0.05
14
0.4
12
0.10
10
0.6
8
0.15
6
0.8
4
0.20
2
1.0
0
0.25
2
CDF - DUniform({1,5,8,11,12})
PMF - DUniform({1,5,8,11,12})
“Error Function”
RISKErf(h)
Parameters:
continuous inverse scale parameter
h
h>0
Domain:
-∞ < x < +∞
continuous
Density and Cumulative Functions:
f (x) =
h −(hx )2
e
π
(
)
F( x ) ≡ Φ 2hx =
1 + erf (hx )
2
where Φ is called the Laplace-Gauss Integral and erf is the Error Function.
Mean:
0
Variance:
1
2h 2
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
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
0.0
-2.0
0.0
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:
⎛x⎞
1
f (x ) =
⎜ ⎟
β (m − 1)! ⎜⎝ β ⎟⎠
m −1
e−x β
Γx β (m )
= 1 − e− x β
F( x ) =
Γ(m )
m− 1
∑
i=0
(x β)i
i!
where Γ is the Gamma Function and Γx is the Incomplete Gamma Function.
Mean:
mβ
Variance:
mβ 2
Skewness:
2
m
Kurtosis:
3+
6
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
7
6
5
4
3
2
1
0
-1
7
6
5
4
3
2
1
0
-1
0.0
Exponential
RISKExpon(β)
Parameters:
β
continuous scale parameter
β>0
Domain:
0 ≤ x < +∞
Density and Cumulative Functions:
f (x) =
e− x β
β
F( x ) = 1 − e − x β
Mean:
β
Variance:
β2
continuous
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
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0.0
-0.5
5.0
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
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:
f (x) =
F( x ) =
⎞
1⎛
1
⎜⎜
⎟
b ⎝ e z + exp(− z ) ⎟⎠
1
e
exp( − z )
Mean:
a − bΓ′(1) ≈ a + .577b
where Γ’(x) is the derivative of the Gamma Function.
Variance:
π2b2
6
where z ≡
(x − a )
b
Skewness:
12 6
π3
ζ (3) ≈ 1.139547
Kurtosis:
5.4
Mode:
a
CDF - ExtValue(0,1)
PDF - ExtValue(0,1)
1.0
0.40
0.9
0.35
0.8
0.30
0.7
0.25
0.6
0.5
0.20
0.4
0.15
0.3
0.10
0.2
0.05
0.1
5
4
3
2
1
0
-1
5
4
3
2
1
0
-1
-2
-2
0.0
0.00
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:
1 ⎛x⎞
⎜ ⎟
f (x) =
β Γ(α ) ⎜⎝ β ⎟⎠
F( x ) =
α −1
e− x β
Γx β (α )
Γ(α )
where Γ is the Gamma Function and Γx is the Incomplete Gamma Function.
Mean:
βα
Variance:
β2α
Skewness:
2
α
Kurtosis:
3+
6
α
Mode:
β(α − 1)
if α ≥ 1
0
if α < 1
CDF - Gamma(4,1)
PDF - Gamma(4,1)
1.0
0.25
0.9
0.8
0.20
0.7
0.6
0.15
0.5
0.4
0.10
0.3
0.2
0.05
0.1
12
10
8
6
4
2
0
12
10
8
6
4
2
0
-2
-2
0.0
0.00
General
RISKGeneral(min, max, {x}, {p})
Parameters:
min
continuous parameter
max
continuous parameter
{x} = {x1, x2, …, xN}
array of continuous parameters
{p} = {p1, p2, …, pN} array of continuous parameters
min < max
min ≤ xi ≤ max
pi ≥ 0
Domain:
min ≤ x ≤ max
continuous
Density and Cumulative Functions:
⎡ x − xi ⎤
f (x) = pi + ⎢
⎥ (p i +1 − p i )
⎣ x i +1 − x i ⎦
for xi ≤ x ≤ xi+1
⎡
(p − p i )(x − x i )⎤
F( x ) = F( x i ) + (x − x i ) ⎢p i + i +1
⎥
2(x i +1 − x i ) ⎦
⎣
for xi ≤ x ≤ xi+1
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
6
5
4
3
2
1
0
0.0
-1
6
5
4
3
2
1
0
-1
0.00
Geometric
RISKGeomet(p)
Parameters:
p
continuous “success” probability
0< p ≤ 1
Domain:
0 ≤ x < +∞
Mass and Cumulative Functions:
f ( x ) = p(1 − p )x
F( x ) = 1 − (1 − p) x +1
Mean:
1
−1
p
Variance:
1− p
p2
discrete integers
Skewness:
(2 − p )
for p < 1
1− p
Not Defined
for p = 1
Kurtosis:
p2
1− p
for p < 1
Not Defined
for p = 1
9+
Mode:
0
CDF - Geomet(.5)
PMF - Geomet(.5)
1.0
0.6
0.9
0.5
0.8
0.7
0.4
0.6
0.5
0.3
0.4
0.2
0.3
0.2
0.1
0.1
7
6
5
4
3
2
1
0
7
6
5
4
3
2
1
0
-1
-1
0.0
0.0
Histogram
RISKHistogrm(min, max, {p})
Parameters:
min
continuous parameter
max
continuous parameter
{p} = {p1, p2, …, pN} array of continuous parameters
min < max *
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) = p i
⎛ x − xi
F( x ) = F( x i ) + p i ⎜⎜
⎝ x i +1 − x i
for xi ≤ x < xi+1
⎞
⎟⎟
⎠
⎛ max − min ⎞
where x i ≡ min + i⎜
⎟
N
⎝
⎠
The {p} array has been normalized to give the histogram unit area.
Mean:
No Closed Form
Variance:
No Closed Form
for xi ≤ x ≤ xi+1
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})
1.0
0.30
0.25
0.8
0.20
0.6
0.15
0.4
0.10
0.2
0.05
6
5
4
3
2
1
0
-1
6
5
4
3
2
1
0
0.0
-1
0.00
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 ⎞
⎟
⎜⎜ ⎟⎟⎜⎜
x ⎠⎝ n − x ⎟⎠
⎝
f (x) =
⎛M⎞
⎜⎜ ⎟⎟
⎝n⎠
x
F( x ) =
∑
i =1
⎛ D ⎞⎛ M − D ⎞
⎟⎟
⎜⎜ ⎟⎟⎜⎜
⎝ x ⎠⎝ n − x ⎠
⎛M⎞
⎜⎜ ⎟⎟
⎝n⎠
Mean:
nD
M
for M > 0
0
for M = 0
Variance:
nD ⎡ (M − D )(M − n )⎤
⎢
(M − 1) ⎥⎦
M2 ⎣
for M>1
0
for M = 1
Skewness:
(M − 2D )(M − 2n )
M−2
M −1
nD(M − D )(M − n )
for M>2, M>D>0, M>n>0
Not Defined
otherwise
Kurtosis:
⎡ M(M + 1) − 6n (M − n ) 3n (M − n )(M + 6) ⎤
M 2 (M − 1)
+
− 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
where x m ≡
(n + 1)(D + 1)
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
6
5
4
3
2
1
0
6
5
4
3
2
1
0
0.0
Integer Uniform
RISKIntUniform(min, max)
Parameters:
min
discrete boundary parameter
max
discrete boundary probability
min < max
Domain:
min ≤ x ≤ max
discrete integers
Mass and Cumulative Functions:
f (x) =
1
max − min + 1
F( x ) =
x − min + 1
max − min + 1
Mean:
min+ max
2
Variance:
∆(∆ + 2)
12
where ∆≡(max-min)
Skewness:
0
Kurtosis:
2
⎛ 9 ⎞ ⎛⎜ n − 7 / 3 ⎞⎟
⎜ ⎟⋅⎜ 2
⎝ 5 ⎠ ⎝ n − 1 ⎟⎠
where n≡(max-min+1)
Mode:
Not uniquely defined
CDF - IntUniform(0,8)
PMF - IntUniform(0,8)
1.0
0.12
0.10
0.8
0.08
0.6
0.06
0.4
0.04
0.2
0.02
9
8
7
6
5
4
3
2
1
0
9
8
7
6
5
4
3
2
1
0
-1
-1
0.0
0.00
Inverse Gaussian
RISKInvGauss(µ, λ)
Parameters:
µ
continuous parameter
µ>0
λ
continuous parameter
λ>0
Domain:
x>0
continuous
Density and Cumulative Functions:
f (x) =
λ
2π x 3
⎡ λ (x − µ ) 2 ⎤
−⎢
⎥
⎢⎣ 2µ 2 x ⎥⎦
e
⎡ λ ⎛ x ⎞⎤
⎡
λ ⎛ x ⎞⎤
⎜⎜ − 1⎟⎟⎥ + e 2λ µ Φ ⎢−
⎜ + 1⎟ ⎥
F( x ) = Φ ⎢
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:
3
µ
λ
Kurtosis:
3 + 15
µ
λ
Mode:
⎡
9µ 2 3µ ⎤
− ⎥
µ ⎢ 1+
2
2λ ⎥
⎢
4λ
⎣
⎦
CDF - InvGauss(1,2)
PDF - InvGauss(1,2)
1.0
1.2
0.9
1.0
0.8
0.7
0.8
0.6
0.5
0.6
0.4
0.4
0.3
0.2
0.2
0.1
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-0.5
0.0
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 ⎞⎞
⎟⎟
sec h 2 ⎜⎜ ⎜⎜
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:
π 2β 2
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
5
4
3
2
1
0
-1
-2
-3
-4
-5
5
4
3
2
1
0
-1
-2
-3
-4
-5
0.0
Log-Logistic
RISKLogLogistic(γ, β, α)
Parameters:
γ
continuous location parameter
β
continuous scale parameter
β>0
α
continuous shape parameter
α>0
Definitions:
θ≡
π
α
Domain:
γ ≤ x < +∞
continuous
Density and Cumulative Functions:
f (x) =
F( x ) =
α t α −1
(
β 1+ tα
)2
1
⎛1⎞
1+ ⎜ ⎟
⎝t⎠
α
with t ≡
Mean:
βθ csc(θ) + γ
for α > 1
x−γ
β
Variance:
[
β 2 θ 2 csc(2θ ) − θ csc 2 (θ )
]
for α > 2
Skewness:
3 csc(3θ) − 6θ csc(2θ) csc(θ) + 2θ 2 csc 3 (θ )
[
θ 2 csc(2θ) − θ csc 2 (θ)
]
3
for α > 3
2
Kurtosis:
4 csc(4θ) − 12θ csc(3θ) csc(θ) + 12θ 2 csc(2θ) csc 2 (θ) − 3θ 3 csc 4 (θ)
[
]
θ 2 csc(2θ) − θ csc (θ )
2
2
for α > 4
Mode:
1
⎡ α − 1⎤ α
γ +β⎢
⎥
⎣ α + 1⎦
for α > 1
γ
for α ≤ 1
CDF - LogLogistic(0,1,5)
PDF - LogLogistic(0,1,5)
1.0
1.4
0.9
1.2
0.8
1.0
0.7
0.6
0.8
0.5
0.6
0.4
0.3
0.4
0.2
0.2
0.1
3.0
2.5
2.0
1.5
1.0
0.5
0.0
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-0.5
0.0
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:
f (x) =
1
x 2 πσ ′
1 ⎡ ln x − µ ′ ⎤
− ⎢
⎥
e 2 ⎣ σ′ ⎦
2
⎛ ln x − µ ′ ⎞
F( x ) = Φ⎜
⎟
⎝ σ′ ⎠
⎡
µ2
⎢
′
with µ ≡ ln
⎢ σ2 + µ2
⎣
⎤
⎥
⎥
⎦
and
⎡ ⎛ σ ⎞2 ⎤
σ ′ ≡ ln ⎢1 + ⎜⎜ ⎟⎟ ⎥
⎢⎣ ⎝ 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:
4
3
⎛σ⎞
with ω ≡ 1 + ⎜⎜ ⎟⎟
⎝µ⎠
2
ω + 2ω + 3ω − 3
2
Mode:
µ4
(σ 2 + µ 2 )3 2
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
6
0.4
5
0.4
4
0.5
3
0.5
2
0.6
1
0.6
6
0.7
5
0.7
4
0.8
3
0.8
2
0.9
1
0.9
0
1.0
-1
1.0
0
CDF - Lognorm(1,1)
-1
PDF - Lognorm(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:
f (x) =
1
x 2 πσ
1 ⎡ ln x − µ ⎤
− ⎢
⎥
e 2⎣ σ ⎦
2
⎛ 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:
e
µ+
σ2
2
Variance:
e 2µ ω(ω − 1)
with ω ≡ e σ
2
Skewness:
(ω + 2)
ω −1
with ω ≡ e σ
2
with ω ≡ e σ
2
Kurtosis:
ω4 + 2ω3 + 3ω2 − 3
Mode:
eµ − σ
2
CDF - Lognorm2(0,1)
PDF - Lognorm2(0,1)
1.0
0.7
0.9
0.6
0.8
0.5
0.7
0.6
0.4
0.5
0.3
0.4
0.3
0.2
0.2
0.1
0.1
12
10
8
6
4
2
0
12
10
8
6
4
2
0
-2
-2
0.0
0.0
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
⎟⎟p (1 − p )x
f ( x ) = ⎜⎜
x
⎝
⎠
F( x ) = p
s
x
⎛ s + i − 1⎞
⎟(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:
3+
6
p2
+
s s(1 − p )
for s > 0, p < 1
Mode:
(bimodal)
z and z + 1
integer z > 0
(unimodal)
0
z<0
(unimodal)
smallest integer greater than z
otherwise
where z ≡
s (1 − p ) − 1
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
9
8
7
6
5
4
3
2
1
0
-1
9
8
7
6
5
4
3
2
1
0
-1
0.0
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:
f (x) =
1
2 πσ
1 ⎛ x −µ ⎞ 2
− ⎜
⎟
e 2⎝ σ ⎠
⎛ x −µ⎞ 1 ⎡ ⎛ x −µ⎞ ⎤
F( x ) ≡ Φ⎜
⎟ + 1⎥
⎟ = ⎢erf ⎜
⎝ 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
3
2
1
0
-1
-2
3
2
1
0
0.0
-1
0.00
-2
0.1
-3
0.05
-3
0.2
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:
f (x) =
θa θ
x θ +1
⎛a⎞
F( x ) = 1 − ⎜ ⎟
⎝x⎠
θ
Mean:
aθ
θ −1
for θ > 1
Variance:
θa 2
(θ − 1)2 (θ − 2)
for θ > 2
Skewness:
2
θ +1 θ − 2
θ−3
θ
for θ > 3
Kurtosis:
(
)
for θ > 4
PDF - Pareto(2,1)
CDF - Pareto(2,1)
3(θ − 2 ) 3θ 2 + θ + 2
θ(θ − 3)(θ − 4 )
Mode:
11
10
0.0
9
0.0
8
0.1
7
0.2
6
0.2
5
0.4
4
0.3
3
0.6
2
0.4
1
0.8
11
0.5
10
1.0
9
0.6
8
1.2
7
0.7
6
1.4
5
0.8
4
1.6
3
0.9
2
1.8
1
1.0
0
2.0
0
a
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:
f (x) =
qb q
(x + b )q +1
F( x ) = 1 −
bq
(x + b )q
Mean:
b
q −1
for q > 1
Variance:
b 2q
(q − 1)2 (q − 2)
for q > 2
Skewness:
⎡ q + 1⎤ q − 2
2⎢
⎥
q
⎣q − 3⎦
for q > 3
Kurtosis:
(
3(q − 2 ) 3q 2 + q + 2
q (q − 3)(q − 4 )
)
for 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
12
10
8
6
4
2
0
-2
12
10
8
6
4
2
0
-2
0.0
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:
f (x) =
1
e −β x
⋅
β Γ(α ) (x β )α +1
F(x) Has No Closed Form
Mean:
β
α −1
for α > 1
Variance:
β2
(α − 1)2 (α − 2)
for α > 2
Skewness:
4 α−2
α−3
for α > 3
Kurtosis:
3(α + 5)(α − 2 )
(α − 3)(α − 4)
for α > 4
Mode:
β
α +1
CDF - Pearson5(3,1)
PDF - Pearson5(3,1)
1.0
2.5
0.9
0.8
2.0
0.7
0.6
1.5
0.5
0.4
1.0
0.3
0.2
0.5
0.1
2.5
2.0
1.5
1.0
0.5
0.0
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-0.5
0.0
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:
f (x) =
(x β ) 1
1
×
β B(α1 , α 2 ) ⎛ x ⎞ α1 + α 2
⎜⎜1 + ⎟⎟
⎝ β⎠
α −1
F(x) Has No Closed Form.
where B is the Beta Function.
Mean:
βα 1
α2 −1
for α2 > 1
Variance:
β 2 α1 (α1 + α 2 − 1)
(α 2 − 1)2 (α 2 − 2)
for α2 > 2
Skewness:
2
⎡ 2α1 + α 2 − 1⎤
α2 − 2
⎢
⎥
α1 (α1 + α 2 − 1) ⎣ α 2 − 3 ⎦
for α2 > 3
Kurtosis:
2
⎤
3 (α 2 − 2 ) ⎡ 2 (α 2 − 1)
+ (α 2 + 5)⎥
⎢
(α 2 − 3)(α 2 − 4) ⎢⎣ α1 (α1 + α 2 − 1)
⎥⎦
for α2 > 4
β(α1 − 1)
α2 +1
for α1 > 1
0
otherwise
Mode:
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
0.3
0.2
0.2
0.1
0.1
0.0
9
8
7
6
5
4
3
2
1
0
-1
9
8
7
6
5
4
3
2
1
0
-1
0.0
Pert (Beta)
RISKPert(min, m.likely, max)
Definitions:
µ≡
min + 4 ⋅ m.likely + max
⎡ µ − min ⎤
α1 ≡ 6 ⎢
6
⎣ max − min ⎥⎦
⎡ max − µ ⎤
α2 ≡ 6 ⎢
⎣ 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:
f (x) =
(x − min )α1 −1 (max− x )α 2 −1
Β(α1 , α 2 )(max − min )α1 + α 2 −1
F( x ) =
B z (α1 , α 2 )
≡ I z (α1 , α 2 )
B(α1 , α 2 )
where B is the Beta Function and Bz is the Incomplete Beta Function.
Mean:
µ≡
min + 4 ⋅ m.likely + max
6
with z ≡
x − min
max − min
Variance:
(µ − min )(max − µ )
7
Skewness:
min + max − 2µ
4
7
(µ − min )(max − µ )
Kurtosis:
3
(α1 + α 2 + 1)(2(α1 + α 2 )2 + α1α 2 (α1 + α 2 − 6))
α1α 2 (α1 + α 2 + 2)(α1 + α 2 + 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
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
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:
f (x) =
λx e −λ
x!
F( x ) = e
−λ
x
∑
n =0
Mean:
λ
Variance:
λ
λn
n!
Skewness:
1
λ
Kurtosis:
3+
1
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
9
8
7
6
5
4
3
2
1
0
-1
9
8
7
6
5
4
3
2
1
0
-1
0.0
Rayleigh
RISKRayleigh(b)
Parameters:
continuous scale parameter
b
b>0
Domain:
0 ≤ x < +∞
continuous
Density and Cumulative Functions:
f (x) =
F( x )
x
b2
1⎛ x ⎞
− ⎜ ⎟
= 1− e 2⎝ b ⎠
Mean:
b
1⎛ x ⎞
− ⎜ ⎟
e 2⎝ b ⎠
π
2
Variance:
π⎞
⎛
b2 ⎜ 2 − ⎟
2⎠
⎝
2
2
Skewness:
2(π − 3) π
(4 − π)3 2
≈ 0.6311
Kurtosis:
32 − 3π 2
(4 − π)2
≈ 3.2451
Mode:
b
CDF - Rayleigh(1)
PDF - Rayleigh(1)
1.0
0.7
0.9
0.6
0.8
0.5
0.7
0.6
0.4
0.5
0.3
0.4
0.3
0.2
0.2
0.1
0.1
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-0.5
0.0
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⎞
ν +1
Γ⎜
⎟
1
⎝ 2 ⎠⎡ ν ⎤ 2
f (x) =
⎢
2⎥
⎛ν⎞
πν
Γ⎜ ⎟ ⎣ ν + x ⎦
⎝2⎠
F( x ) =
1⎡
⎛ 1 ν ⎞⎤
1 + I s ⎜ , ⎟⎥
⎢
2⎣
⎝ 2 2 ⎠⎦
with s ≡
x2
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
ν−2
for ν > 2
Skewness:
for ν > 3*
0
*even though the skewness is not defined for ν ≤ 3, the distribution is still symmetric about 0.
Kurtosis:
⎛ν −2⎞
3⎜
⎟
⎝ν −4⎠
for 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
5
4
3
2
1
0
-1
-2
-3
-4
-5
5
4
3
2
1
0
-1
-2
-3
-4
-5
0.0
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:
f (x) =
2(x − min )
(m.likely − min)(max − min)
min ≤ x ≤ m.likely
f (x) =
2(max − x )
(max − m.likely)(max − min)
m.likely ≤ x ≤ max
(
x − min )2
F( x ) =
(m.likely − min )(max − min )
min ≤ x ≤ m.likely
F( x ) = 1 −
(max − x )2
(max − m.likely)(max − min )
Mean:
min + m.likely + max
3
m.likely ≤ x ≤ max
Variance:
min 2 + m.likely2 + max 2 − (max )(m.likely) − (m.likely)(min ) − (max )(min )
18
Skewness:
(
)
2 2 f f2 −9
32
5
f2 +3
(
)
where f ≡
2( m.likely − min)
−1
max − min
Kurtosis:
2.4
Mode:
m.likely
PDF - Triang(0,3,5)
CDF - Triang(0,3,5)
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
6
5
4
3
2
1
0
-1
6
5
4
3
2
1
0
0.0
-1
0.00
Uniform
RISKUniform(min, max)
Parameters:
min
continuous boundary parameter
max
continuous boundary parameter
min < max *
*min = max is supported for modeling convenience, but gives a degenerate distribution.
Domain:
min ≤ x ≤ max
Density and Cumulative Functions:
f (x) =
1
max − min
F( x ) =
x − min
max − min
Mean:
max− min
2
Variance:
(max− min )2
12
continuous
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
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
1.2
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
0.0
Weibull
RISKWeibull(α, β)
Parameters:
α
continuous shape parameter
α>0
β
continuous scale parameter
β>0
Domain:
0 ≤ x < +∞
Density and Cumulative Functions:
f (x) =
αx α −1 − (x β )α
e
βα
α
F( x ) = 1 − e − ( x β )
Mean:
1⎞
⎛
β Γ⎜1 + ⎟
⎝ α⎠
where Γ is the Gamma Function.
Variance:
⎡ ⎛
2⎞
1 ⎞⎤
⎛
β 2 ⎢Γ⎜1 + ⎟ − Γ 2 ⎜1 + ⎟⎥
⎝ α ⎠⎦
⎣ ⎝ α⎠
where Γ is the Gamma Function.
continuous
Skewness:
3⎞
2⎞ ⎛
1⎞
1⎞
⎛
⎛
⎛
Γ⎜1 + ⎟ − 3Γ⎜1 + ⎟Γ⎜1 + ⎟ + 2Γ 3 ⎜1 + ⎟
⎝ α⎠
⎝ α⎠ ⎝ α⎠
⎝ α⎠
⎡ ⎛
2⎞
1 ⎞⎤
2⎛
⎢Γ⎜1 + α ⎟ − Γ ⎜1 + α ⎟ ⎥
⎠
⎝
⎠⎦
⎣ ⎝
32
where Γ is the Gamma Function.
Kurtosis:
4⎞
3⎞ ⎛
1⎞
2⎞ ⎛
1⎞
1⎞
⎛
⎛
⎛
⎛
Γ⎜1 + ⎟ − 4Γ⎜1 + ⎟Γ⎜1 + ⎟ + 6Γ⎜1 + ⎟Γ 2 ⎜1 + ⎟ − 3Γ 4 ⎜1 + ⎟
⎝ α⎠
⎝ α⎠ ⎝ α⎠
⎝ α⎠ ⎝ α⎠
⎝ α⎠
⎡ ⎛
2⎞
1 ⎞⎤
2⎛
⎢Γ⎜1 + α ⎟ − Γ ⎜1 + α ⎟⎥
⎠
⎝
⎠⎦
⎣ ⎝
2
where Γ is the Gamma Function.
Mode:
1α
⎛ 1⎞
β⎜1 − ⎟
⎝ α⎠
for α >1
0
for α ≤ 1
CDF - Weibull(2,1)
PDF - 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
2.5
2.0
1.5
1.0
0.5
0.0
2.5
2.0
1.5
1.0
0.5
0.0
0.0
0.0
-0.5
0.1
-0.5
0.2
0.1
```
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