funciones de probabilidad
DESCRIPTION
BinomialPoissonNormalLognormalTRANSCRIPT
A Concise Summary of @RISK Probability Distribution Functions
Copyright 2002 Palisade Corporation
All Rights Reserved
Beta RISKBeta(α1, α2)
Parameters: αααα1 continuous shape parameter α1 > 0 αααα2 continuous shape parameter α2 > 0 Domain: 0 ≤ x ≤ 1 continuous
Density and Cumulative Distribution Functions:
( )),(
x1x)x(f21
11 21
ααΒ−=
−α−α
( ) ( )21x21
21x ,I),(B
,B)x(F αα≡
αααα
=
where B is the Beta Function and Bx is the Incomplete Beta Function. Mean:
21
1α+α
α
Variance:
( ) ( )1212
21
21
+α+αα+ααα
Skewness:
21
21
21
12 12
2αα
+α+α+α+α
α−α
Kurtosis:
( ) ( ) ( )( )
( )( )32621
3212121
21212
2121+α+α+α+ααα
−α+ααα+α+α+α+α
Mode:
21
21
1−α+α
−α 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)
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0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
-0.2 0.0
0.2
0.4
0.6
0.8
1.0
1.2
CDF - Beta(2,3)
0.0
0.2
0.4
0.6
0.8
1.0
-0.2 0.0
0.2
0.4
0.6
0.8
1.0
1.2
Beta (Generalized) RISKBetaGeneral(α1, α2, min, max)
Parameters: αααα1 continuous shape parameter α1 > 0 αααα2 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:
( ) ( )( ) 12121
1211
minmax),(xmaxminx)x(f −α+α
−α−α
−ααΒ−−=
( ) ( )21z21
21z ,I),(B
,B)x(F αα≡
αααα
= with minmax
minxz−
−≡
where B is the Beta Function and Bz is the Incomplete Beta Function. Mean:
( )minmaxmin21
1 −α+α
α+
Variance:
( ) ( )( ) 2
212
21
21 minmax1
−+α+αα+α
αα
Skewness:
21
21
21
12 12
2αα
+α+α+α+α
α−α
Kurtosis:
( ) ( ) ( )( )
( )( )32621
3212121
21212
2121+α+α+α+ααα
−α+ααα+α+α+α+α
Mode:
( )minmax2
1min
21
1 −−α+α
−α+ 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)
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0.05
0.10
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0.25
0.30
0.35
0.40
-1 0 1 2 3 4 5 6
CDF - BetaGeneral(2,3,0,5)
0.0
0.2
0.4
0.6
0.8
1.0
-1 0 1 2 3 4 5 6
Beta (Subjective) RISKBetaSubj(min, m.likely, mean, max)
Definitions:
2maxminmid +≡
( )( )( )( )minmaxlikely.mmean
likely.mmidminmean21 −−−−≡α
minmeanmeanmax
12 −−α≡α
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:
( ) ( )( ) 12121
1211
minmax),(xmaxminx)x(f −α+α
−α−α
−ααΒ−−=
( ) ( )21z21
21z ,I),(B
,B)x(F αα≡
αααα
= with minmax
minxz−
−≡
where B is the Beta Function and Bz is the Incomplete Beta Function.
Mean:
mean Variance:
( )( )( )likely.m3meanmid2
likely.mmeanmeanmaxminmean⋅−+⋅
−−−
Skewness:
( ) ( )( )( )( )meanmaxminmean
likely.m3meanmid2likely.mmeanlikely.m2midmean
meanmid2−−
⋅−+⋅−⋅−+
−
Kurtosis:
( ) ( ) ( )( )
( )( )32621
3212121
21212
2121+α+α+α+ααα
−α+ααα+α+α+α+α
Mode: m.likely
PDF - BetaSubj(0,1,2,5)
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0.30
-1 0 1 2 3 4 5 6
CDF - BetaSubj(0,1,2,5)
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0.2
0.4
0.6
0.8
1.0
-1 0 1 2 3 4 5 6
Binomial RISKBinomial(n, p)
Parameters: n discrete “count” parameter n > 0 p continuous “success” probability 0 < p < 1 Domain: 0 ≤ x ≤ n discrete
Mass and Cumulative Functions:
( ) xnx p1pxn
)x(f −−
=
inix
0i
)p1(pin
)x(F −
=
−
=∑
Mean:
np
Variance:
( )p1np −
Skewness:
( )( )p1np
p21−
−
Kurtosis:
( )p1np1
n63
−+−
Mode:
(bimodal) ( ) 11np −+ and ( )1np + if ( )1np + is integral (unimodal) largest integer less than ( )1np + otherwise
PMF - Binomial(8,.4)
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0.20
0.25
0.30
-1 0 1 2 3 4 5 6 7 8 9
CDF - Binomial(8,.4)
0.0
0.2
0.4
0.6
0.8
1.0
-1 0 1 2 3 4 5 6 7 8 9
Chi-Squared RISKChiSq(ν)
Parameters: νννν discrete shape parameter ν > 0 Domain: 0 ≤ x ≤ +∞ continuous
Density and Cumulative Functions:
( )
( ) 122x2 xe
221)x(f −ν−
ν νΓ=
( )
( )22
)x(F 2x
νΓ
νΓ=
where Γ is the Gamma Function, and Γx is the Incomplete Gamma Function. Mean:
ν Variance:
2ν
Skewness:
ν8
Kurtosis:
ν+123
Mode:
ν-2 if ν ≥ 2 0 if ν = 1
PDF - ChiSq(5)
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0.02
0.04
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0.08
0.10
0.12
0.14
0.16
0.18
-2 0 2 4 6 8 10 12 14 16
CDF - ChiSq(5)
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0.1
0.2
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0.4
0.5
0.6
0.7
0.8
0.9
1.0
-2 0 2 4 6 8 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:
i1i
i1ixxpp
)x(f−−
=+
+ for xi ≤ x < xi+1
( )
−
−−+=
++
i1i
ii1ii xx
xxppp)x(F 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
CDF - Cumul({1,2,3,4},{.2,.3,.7,.8})
0.0
0.2
0.4
0.6
0.8
1.0
-1 0 1 2 3 4 5 6
PDF - Cumul({1,2,3,4},{.2,.3,.7,.8})
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0.45
-1 0 1 2 3 4 5 6
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:
i1i
1iixx
pp)x(f
−−
=+
+ for xi ≤ x < xi+1
( )
−
−−+−=
++
i1i
i1iii xx
xxppp1)x(F 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({1,2,3,4},{.2,.3,.7,.8})
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0.45
-1 0 1 2 3 4 5 6
CDF - CumulD({1,2,3,4},{.2,.3,.7,.8})
0.0
0.2
0.4
0.6
0.8
1.0
-1 0 1 2 3 4 5 6
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: ip)x(f = for ixx = 0)x(f = for }x{x ∉
0)x(F = for x < x1
∑=
=s
1iip)x(F for xs ≤ x < xs+1, s < N
1)x(F = for x ≥ xN
With the assumptions: 1. The arrays are ordered from left to right 2. The p array is normalized to 1. Mean:
µ≡∑=
i
N
1iipx
Variance:
Vp)x( i2
N
1ii ≡µ−∑
=
Skewness:
i3
N
1ii23 p)x(
V1 ∑
=
µ−
Kurtosis:
i4
N
1ii2 p)x(
V1 ∑
=
µ−
Mode: The x-value corresponding to the highest p-value.
CDF - Discrete({1,2,3,4},{2,1,2,1})
0.0
0.2
0.4
0.6
0.8
1.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
PMF - Discrete({1,2,3,4},{2,1,2,1})
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0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Discrete Uniform RISKDUniform({x})
Parameters: {x} = {x1, x1, …, xN} array of continuous parameters Domain: }x{x ∈ discrete
Mass and Cumulative Functions:
N1)x(f = for }x{x ∈
0)x(f = for }x{x ∉
0)x(F = for x < x1
Ni)x(F = for xi ≤ x < xi+1
1)x(F = for x ≥ xN
assuming the {x} array is ordered. Mean:
µ≡∑=
N
1iix
N1
Variance:
V)x(N1 2
N
1ii ≡µ−∑
=
Skewness:
3N
1ii23 )x(
NV1 ∑
=
µ−
Kurtosis:
4N
1ii2 )x(
NV1 ∑
=
µ−
Mode: Not uniquely defined
PMF - DUniform({1,5,8,11,12})
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0.10
0.15
0.20
0.25
0 2 4 6 8 10 12 14
CDF - DUniform({1,5,8,11,12})
0.0
0.2
0.4
0.6
0.8
1.0
0 2 4 6 8 10 12 14
“Error Function” RISKErf(h)
Parameters: h continuous inverse scale parameter h > 0 Domain:
-∞ ≤ x ≤ +∞ continuous
Density and Cumulative Functions:
( )2hxeh)x(f −π
=
( )hx2)x(F Φ= where Φ is the Error Function. Mean:
0 Variance:
2h21
Skewness:
0 Kurtosis:
3 Mode:
0
PDF - Erf(1)
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0.5
0.6
-2.0
-1.5
-1.0
-0.5 0.0
0.5
1.0
1.5
2.0
CDF - Erf(1)
0.0
0.1
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0.5
0.6
0.7
0.8
0.9
1.0
-2.0
-1.5
-1.0
-0.5 0.0
0.5
1.0
1.5
2.0
Erlang RISKErlang(m,β)
Parameters: m integral shape parameter m > 0
ββββ continuous scale parameter β > 0
Domain:
0 ≤ x < +∞ continuous Density and Cumulative Functions:
β−−α
β−β
= x1ex
)!1m(1)x(f
( )
( )( )∑
−
=
β−β β−=
Γ
Γ=
1m
0i
ixx
!ix
e1m
m)x(F
where Γ is the Gamma Function and Γx is the Incomplete Gamma Function. Mean: βm Variance:
2mβ
Skewness:
m2
Kurtosis:
m63 +
Mode: ( )1m −β
PDF - Erlang(2,1)
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-1 0 1 2 3 4 5 6 7
CDF - Erlang(2,1)
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0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
-1 0 1 2 3 4 5 6 7
Exponential RISKExpon(β)
Parameters: ββββ continuous scale parameter β > 0
Domain: 0 ≤ x < +∞ continuous
Density and Cumulative Functions:
β
=β−xe)x(f
β−−= xe1)x(F
Mean:
β Variance:
β2
Skewness:
2 Kurtosis:
9 Mode:
0
PDF - Expon(1)
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0.8
1.0
1.2
-0.5 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
CDF - Expon(1)
0.0
0.1
0.2
0.3
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0.5
0.6
0.7
0.8
0.9
1.0-0
.5 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Extreme Value RISKExtValue(a, b)
Parameters: a continuous location parameter
b continuous scale parameter b > 0 Domain:
-∞ ≤ x ≤ +∞ continuous
Density and Cumulative Functions:
= −+ )zexp(ze
1b1)x(f
)zexp(e
1)x(F−
= where ( )b
axz −≡
Mean:
( ) b577.a1ba +≈Γ′− where Γ’(x) is the derivative of the Gamma Function. Variance:
6b22π
Skewness: 1.139547
Kurtosis:
5.4
Mode:
a
PDF - ExtValue(0,1)
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-2 -1 0 1 2 3 4 5
CDF - ExtValue(0,1)
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0.5
0.6
0.7
0.8
0.9
1.0
-2 -1 0 1 2 3 4 5
Gamma RISKGamma(α, β)
Parameters: αααα continuous shape parameter α > 0
ββββ continuous scale parameter β > 0
Domain:
0 < x < +∞ continuous
Density and Cumulative Functions:
( )β−
−α
βαΓβ
= x1ex1)x(f
( )
( )αΓαΓ
= βx)x(F
where Γ is the Gamma Function and Γx is the Incomplete Gamma Function. Mean: βα Variance:
αβ2
Skewness:
α2
Kurtosis:
α
+ 63
Mode: ( )1−αβ if α ≥ 1 Not Defined if α < 1
CDF - Gamma(4,1)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
-2 0 2 4 6 8 10 12
PDF - Gamma(4,1)
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-2 0 2 4 6 8 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:
( )i1ii1i
ii pp
xxxx
p)x(f −
−
−+= +
+ for xi ≤ x ≤ xi+1
( ) ( )( )( )
−
−−+−+=
+
+
i1i
ii1iiii xx2
xxpppxx)x(F)x(F 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})
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0.35
-1 0 1 2 3 4 5 6
CDF - General(0,5,{1,2,3,4},{2,1,2,1})
0.0
0.2
0.4
0.6
0.8
1.0-1 0 1 2 3 4 5 6
Geometric RISKGeomet(p)
Parameters: p continuous “success” probability 0< p < 1 Domain: 0 ≤ x < +∞ discrete
Mass and Cumulative Functions: ( )xp1p)x(f −=
1x)p1(1)x(F +−−=
Mean:
1p1 −
Variance:
2pp1−
Skewness:
( )p1p2
−−
Kurtosis:
( )2p1p9
−+
Mode:
0
PMF - Geomet(.5)
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0.2
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0.5
0.6
-1 0 1 2 3 4 5 6 7
CDF - Geomet(.5)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0-1 0 1 2 3 4 5 6 7
Histogram RISKHistogrm(min, max, {p})
Parameters: min continuous parameter min < max max continuous parameter {p} = {p1, p2, …, pN} array of continuous parameters pi ≥ 0 Domain: min ≤ x ≤ max continuous
Density and Cumulative Functions: ip)x(f = for xi ≤ x < xi+1
−
−+=
+ i1i
iii xx
xxp)x(F)x(F for xi ≤ x ≤ xi+1
where
−+≡
Nminmaximinxi
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})
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CDF - Histogrm(0,5,{6,5,3,4,5})
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1.0
-1 0 1 2 3 4 5 6
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
Mass and Cumulative Functions:
−−
=
nM
xnDM
xD
)x(f
∑=
−−
=x
1inM
xnDM
xD
)x(F
Mean:
MnD
Variance:
( )( )( )
−
−−1M
nMDMMnD
2 for M>1
Skewness:
( )( )( ) )nM(DMnD
1M2M
n2MD2M−−
−−
−− for M>2
Mode:
(bimodal) xm and xm-1 if xm is integral
(unimodal) biggest integer less than xm otherwise
where ( )( )2M
1D1nx m +++≡
PMF - HyperGeo(6,5,10)
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0.45
0.50
0 1 2 3 4 5 6
CDF - HyperGeo(6,5,10)
0.0
0.2
0.4
0.6
0.8
1.0
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
Mass and Cumulative Functions:
1minmax
1)x(f+−
=
1minmax1minx)x(F+−
+−=
Mean:
2maxmin+
Variance:
+−
− 1
2minmax
6minmax
Skewness:
0 Mode:
Not uniquely defined
CDF - IntUniform(0,8)
0.0
0.2
0.4
0.6
0.8
1.0
-1 0 1 2 3 4 5 6 7 8 9
PMF - IntUniform(0,8)
0.00
0.02
0.04
0.06
0.08
0.10
0.12
-1 0 1 2 3 4 5 6 7 8 9
Inverse Gaussian RISKInvGauss(µ, λ)
Parameters:
µµµµ continuous parameter µ > 0
λλλλ continous parameter λ > 0 Domain:
x > 0 continuous
Density and Cumulative Functions:
( )
µ
µ−λ−
πλ= x2
x
3
2
2
ex2
)x(f
+µ
λ−Φ+
−µ
λΦ= µλ 1xx
e1xx
)x(F 2
where Φ is the Error Function. Mean: µ Variance:
λ
µ3
Skewness:
λµ3
Kurtosis:
λµ+153
Mode:
λµ−
λµ+µ
23
491 2
2
PDF - InvGauss(1,2)
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0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
CDF - InvGauss(1,2)
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0.1
0.2
0.3
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0.6
0.7
0.8
0.9
1.0
-0.5 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Logistic RISKLogistic(α, β)
Parameters: αααα continuous location parameter
ββββ continuous scale parameter β > 0
Domain: -∞ ≤ x ≤ +∞ continuous
Density and Cumulative Functions:
β
β
α−
=4
x21hsec
)x(f
2
2
x21tanh1
)x(F
β
α−+=
where “sech” is the Hyperbolic Secant Function and “tanh” is the Hyperbolic Tangent Function. Mean:
α Variance:
3
22βπ
Skewness:
0 Kurtosis:
4.2 Mode:
α
PDF - Logistic(0,1)
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CDF - Logistic(0,1)
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Log-Logistic RISKLogLogistic(γ, β, α)
Parameters:
γγγγ continuous location parameter
ββββ continous scale parameter β > 0 αααα continuous shape parameter α > 0
Definitions:
απ≡θ
Domain:
γ ≤ x ≤ +∞ continuous
Density and Cumulative Functions:
( )21
t1
t)x(fα
−α
+β
α=
α
+
=
t11
1)x(F with β
γ−≡ xt
Mean:
( ) γ+θβθ csc for α > 1
Variance:
( ) ( )[ ]θθ−θθβ 22 csc2csc2 for α > 2
Skewness:
( ) ( ) ( ) ( )
( ) ( )[ ] 23
2
32
csc2csc2
csc2csc2csc63csc3
θθ−θθ
θθ+θθθ−θ for α > 3
Mode:
α
+α−αβ+γ
1
11 for α > 1
0 for α ≤ 1
PDF - LogLogistic(0,1,5)
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CDF - LogLogistic(0,1,5)
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Lognormal (Format 1) RISKLognorm(µ, σ)
Parameters:
µµµµ continuous parameter µ > 0
σσσσ continous parameter σ > 0
Domain:
0 ≤ x ≤ +∞ continuous
Density and Cumulative Functions:
2xln
21
e2x1)x(f
σ′µ′−−
σ′π=
σ′µ′−Φ= xln)x(F
with
µ+σ
µ≡µ′22
2ln and
+≡′
2
1lnµσσ
where Φ is the Error Function. Mean: µ Variance:
2σ
Skewness:
µσ+
µσ 3
3
Kurtosis:
332 234 −ω+ω+ω with
µσ+≡ω 1
Mode:
( ) 2322
4
µ+σ
µ
PDF - Lognorm(1,1)
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CDF - Lognorm(1,1)
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Lognormal (Format 2) RISKLognorm2(µ, σ)
Parameters:
µµµµ continuous parameter
σσσσ continous parameter σ > 0
Domain:
0 ≤ x ≤ +∞ continuous
Density and Cumulative Functions:
2xln21
e2x1)x(f
σµ−−
σπ=
σµ−Φ= xln)x(F
where Φ is the Error Function. Mean:
2
2
eσ+µ
Variance:
( )1e2 −ωωµ with 2
eσ≡ω
Skewness:
( ) 12 −ω+ω with 2
eσ≡ω Kurtosis:
332 234 −ω+ω+ω with 2
eσ≡ω Mode:
2
e σ−µ
PDF - Lognorm2(0,1)
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CDF - Lognorm2(0,1)
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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 ≤ n discrete
Density and Cumulative Functions:
( )xs p1px
1xs)x(f −
−+=
ix
0i
s )p1(i
1isp)x(F −
−+= ∑
=
Where ( ) is the Binomial Coefficient. Mean:
( )p
p1s −
Variance:
( )2p
p1s −
Skewness:
( )p1sp2−
−
Kurtosis:
( )p1sp
s63
2
−++
Mode:
(bimodal) z and z + 1 integer z > 0
(unimodal) 0 z < 0 (unimodal) smallest integer greater than z otherwise
where ( )p
1p1sz −−≡
PDF - NegBin(3,.6)
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CDF - NegBin(3,.6)
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0.9
1.0
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Normal RISKNormal(µ, σ)
Parameters: µµµµ continuous location parameter
σσσσ continuous scale parameter σ > 0
Domain:
-∞ ≤ x ≤ +∞ continuous
Density and Cumulative Functions:
2x21
e21)x(f
σµ−−
σπ=
σµ−Φ= x)x(F
where Φ is the Error Function. Mean:
µ Variance:
σ2
Skewness:
0 Kurtosis:
3 Mode:
µ
PDF - Normal(0,1)
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CDF - Normal(0,1)
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1.0
-3 -2 -1 0 1 2 3
Pareto (First Kind) Pareto(θ, a)
Parameters: θθθθ continuous shape parameter θ > 0 a continuous scale parameter a > 0
Domain: a ≤ x ≤ +∞ continuous
Density and Cumulative Functions:
1x
a)x(f+θ
θθ=
θ
−=
xa1)x(F
Mean:
1a−θθ for θ > 1
Variance:
( ) ( )21
a2
2
−θ−θ
θ for θ > 2
Skewness:
θ−θ
−θ+θ 2
312 for θ > 3
Kurtosis:
( )( )( )( )43
2323 2
−θ−θθ+θ+θ−θ for θ > 4
Mode:
a
PDF - Pareto(2,1)
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CDF - Pareto(2,1)
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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:
( ) 1q
q
bxqb)x(f ++
=
bxb1)x(F+
−=
Mean:
1qb−
for q > 1
Variance:
( ) ( )2q1q
qb2
2
−− for q > 2
Skewness:
−+−
3q1q
q2q2 for q > 3
Mode: 0
PDF - Pareto2(3,3)
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CDF - Pareto2(3,3)
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Pearson Type V RISKPearson5(α, β)
Parameters:
αααα continuous shape parameter α > 0
ββββ continous scale parameter β > 0
Domain:
0 ≤ x < +∞ continuous
Density and Cumulative Functions:
( ) ( ) 1
x
xe1)x(f +α
β−
β⋅
αΓβ=
F(x) Has No Closed Form Mean:
1−α
β for α > 1
Variance:
( ) ( )21 2
2
−α−α
β for α > 2
Skewness:
3
24−α−α for α > 3
Kurtosis:
( )( )( )( )43
253−α−α−α+α for α > 4
Mode:
1+α
β
PDF - Pearson5(3,1)
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CDF - Pearson5(3,1)
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Pearson Type VI RISKPearson6(α1, α2, β)
Parameters:
αααα1 continuous shape parameter α1 > 0 αααα2 continuous shape parameter α2 > 0
ββββ continuous scale parameter β > 0
Domain:
0 ≤ x < +∞ continuous
Density and Cumulative Functions:
( )( )
21
1
x1
x,B
1)x(f1
21α+α
−α
β
+
β×ααβ
=
F(x) Has No Closed Form.
where B is the Beta Function. Mean:
12
1
−αβα
for α2 > 1
Variance:
( )( ) ( )21
1
22
2
2112
−α−α−α+ααβ
for α2 > 2
Skewness:
( )
−α
−α+α−α+αα
−α3
121
22
2
21
211
2 for α2 > 3
Kurtosis:
( )( )( )
( )( ) ( )
+α+
−α+αα−α
−α−α−α
51
1243
232
211
22
22
2 for α2 > 4
Mode:
( )11
2
1
+α−αβ
for α1 > 1
0 otherwise
PDF - Pearson6(3,3,1)
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CDF - Pearson6(3,3,1)
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Pert (Beta) RISKPert(min, m.likely, max)
Definitions:
6maxlikely.m4min +⋅+≡µ
−−µ≡α
minmaxmin61
−µ−≡α
minmaxmax62
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:
( ) ( )( ) 12121
1211
minmax),(xmaxminx)x(f −α+α
−α−α
−ααΒ−−=
( ) ( )21z21
21z ,I),(B
,B)x(F αα≡
αααα
= with minmax
minxz−
−≡
where B is the Beta Function and Bz is the Incomplete Beta Function. Mean:
6maxlikely.m4min +⋅+≡µ
Variance:
( )( )7
maxmin µ−−µ
Skewness:
( )( )µ−−µµ−+
maxmin7
42maxmin
Kurtosis:
( ) ( ) ( )( )( )( )32
6213
212121
21212
2121+α+α+α+ααα
−α+ααα+α+α+α+α
Mode: m.likely
PDF - Pert(0,1,3)
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CDF - Pert(0,1,3)
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Poisson RISKPoisson(λ)
Parameters: λλλλ mean number of successes continuous λ > 0 Domain: 0 ≤ x < +∞ discrete
Mass and Cumulative Functions:
!xe)x(f
x λ−λ=
∑=
λ− λ=x
0n
n
!ne)x(F
Mean:
λ Variance:
λ
Skewness:
λ1
Kurtosis:
λ+ 13
Mode:
(bimodal) λ and λ+1 (bimodal) if λ is an integer
(unimodal) largest integer less than λ otherwise
PMF - Poisson(3)
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CDF - Poisson(3)
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Rayleigh RISKRayleigh(b)
Parameters: b continuous scale parameter b > 0 Domain:
0 ≤ x < +∞ continuous
Density and Cumulative Functions:
2
bx
21
2e
bx)x(f
−
=
2
bx
21
e1)x(F
−
−= Mean:
2b π
Variance:
π−
22b2
Skewness:
( )( )
6311.04
3223
≈π−
π−π
Kurtosis:
( )2451.3
4332
2
2≈
π−
π−
Mode:
b
PDF - Rayleigh(1)
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CDF - Rayleigh(1)
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Student’s “t” RISKStudent(ν)
Parameters: νννν the degrees of freedom integer ν > 0
Domain:
-∞ ≤ x ≤ +∞ continuous
Density and Cumulative Functions:
21
2x2
21
1)x(f
+ν
+νν
νΓ
+νΓ
πν=
ν+=
2,
21I1
21)x(F s with 2
2
xxs+ν
≡
where Γ is the Gamma Function and Ix is the Incomplete Beta Function. Mean:
0 for ν > 1* *even though the mean is not well defined for ν = 1, the distribution is still symmetrical about 0. Variance:
2−νν for ν > 2
Skewness:
0 for ν > 3* *even though the skewness is not well defined for ν ≤ 3, the distribution is still symmetric about 0. Kurtosis:
−ν−ν
423 for ν > 4
Mode:
0
PDF - Student(3)
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CDF - Student(3)
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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 Domain: min ≤ x ≤ max continuous
Density and Cumulative Functions:
( )min)min)(maxlikely.m(
minx2)x(f−−
−= min ≤ x ≤ m.likely
( )
min))(maxlikely.m(maxxmax2)x(f
−−−= m.likely ≤ x ≤ max
( )( )( )minmaxminlikely.m
minx)x(F2
−−−= min ≤ x ≤ m.likely
( )( )( )minmaxlikely.mmax
xmax1)x(F2
−−−−= m.likely ≤ x ≤ max
Mean:
3maxlikely.mmax ++
Variance:
( )( ) ( )( ) ( )( )18
minmaxminlikely.mlikely.mmaxmaxlikely.mmax 222 −−−++
Skewness:
( )( ) 232
2
3f
9ff5
22
+
− where ( ) 1minmax
minlikely.m2f −−
−≡
Kurtosis: 2.4 Mode:
m.likely
PDF - Triang(0,3,5)
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CDF - Triang(0,3,5)
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Uniform RISKUniform(min, max)
Parameters: min continuous boundary parameter min < max max continuous boundary parameter Domain: min ≤ x ≤ max continuous
Density and Cumulative Functions:
minmax
1)x(f−
=
minmaxminx)x(F−
−=
Mean:
2minmax−
Variance:
( )12
minmax 2−
Skewness:
0 Kurtosis:
1.8 Mode:
Not uniquely defined
CDF - Uniform(0,1)
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PDF - Uniform(0,1)
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Weibull RISKWeibull(α, β)
Parameters: αααα continuous shape parameter α > 0
ββββ continuous scale parameter β > 0
Domain:
0 ≤ x < +∞ continuous
Density and Cumulative Functions:
( )αβ−α
−α
βα= x
1ex)x(f
( )αβ−−= xe1)x(F Mean:
α+Γ 11b
where Γ is the Gamma Function. Variance:
α+Γ−
α+Γβ
11
21 22
where Γ is the Gamma Function.
Skewness:
232
3
1121
1121121331
α+Γ−
α+Γ
α+Γ+
α+Γ
α+Γ+
α+Γ
where Γ is the Gamma Function. Mode:
α
α−β
111 for α >1
0 for α ≤ 1
PDF - Weibull(2,1)
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CDF - Weibull(2,1)
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