Post on 03-Apr-2018
7/28/2019 Much as Much as Dis Tri Buci Ones
1/75
A Concise Summary of @RISK ProbabilityDistribution Functions
Copyright 2002 Palisade Corporation
All Rights Reserved
7/28/2019 Much as Much as Dis Tri Buci Ones
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Skewness:
21
21
21
12 1
22
++
++
Kurtosis:
( ) ( ) ( )( )( )( )32
6213
212121
21212
2121
++++
+++++
Mode:
2
1
21
1
+
1>1, 2>1
0 11
1 11, 21, 2=1
PDF - Beta(2,3)
0.0
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
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Beta (Generalized)
RISKBetaGeneral(1, 2, min, max)
Parameters:
1 continuous shape parameter 1> 0
2 continuous shape parameter 2> 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
+
=
( )( )21z
21
21z ,I),(B
,B)x(F
= with
minmax
minxz
where B is theBetaFunction and Bz is theIncompleteBetaFunction.
Mean:
( )minmaxmin
21
1 +
+
Variance:
( ) ( )( ) 2
212
21
21 minmax1
+++
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Skewness:
21
21
21
12 1
22
++
++
Kurtosis:
( ) ( ) ( )( )( )( )32
6213
212121
21212
2121
++++
+++++
Mode:
( )minmax2
1min
21
1 +
+ 1>1, 2>1
min 11
max 11, 21, 2=1
PDF - BetaGeneral(2,3,0,5)
0.00
0.05
0.10
0.15
0.20
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
7/28/2019 Much as Much as Dis Tri Buci Ones
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Beta (Subjective)
RISKBetaSubj(min, m.likely, mean, max)
Definitions:
2
maxminmid
+
( )( )( )( )minmaxlikely.mmean
likely.mmidminmean21
minmean
meanmax12
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 < meanmean = mid if m.likely = mean
Domain:
min x max continuous
Density and Cumulative Distribution Functions:
( ) ( )
( ) 12121
1211
minmax),(
xmaxminx)x(f +
=
( )( )21z
21
21z ,I),(B
,B)x(F
= with
minmax
minxz
where B is theBeta Function and Bz is theIncomplete Beta Function.
7/28/2019 Much as Much as Dis Tri Buci Ones
7/75
Mean:
mean
Variance:
( )( )( )
likely.m3meanmid2
likely.mmeanmeanmaxminmean
+
Skewness:
( ) ( )( )
( )( )meanmaxminmean
likely.m3meanmid2likely.mmean
likely.m2midmean
meanmid2
+
+
Kurtosis:
( ) ( ) ( )
( )( )32
6213
212121
21212
2121
++++
+++++
Mode:
m.likely
PDF - BetaSubj(0,1,2,5)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
-1 0 1 2 3 4 5 6
CDF - BetaSubj(0,1,2,5)
0.0
0.2
0.4
0.6
0.8
1.0
-1 0 1 2 3 4 5 6
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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 p1px
n)x(f
=
inix
0i
)p1(pi
n)x(F
=
=
Mean:
np
Variance:
( )p1np
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Skewness:
( )
( )p1np
p21
Kurtosis:
( )p1np1
n
63
+
Mode:
(bimodal) ( ) 11np + and ( )1np + if ( )1np + is integral
(unimodal) largest integer less than ( )1np + otherwise
PMF - Binomial(8,.4)
0.00
0.05
0.10
0.15
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
7/28/2019 Much as Much as Dis Tri Buci Ones
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Chi-Squared
RISKChiSq()
Parameters:
discrete shape parameter > 0
Domain:
0 x + continuous
Density and Cumulative Functions:
( )( ) 122x
2xe
22
1)x(f
=
( )
( )2
2)x(F
2x
=
where is the Gamma Function, and x is theIncomplete Gamma Function.
Mean:
Variance:
2
7/28/2019 Much as Much as Dis Tri Buci Ones
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Skewness:
8
Kurtosis:
+12
3
Mode:
-2 if 2
0 if = 1
PDF - ChiSq(5)
0.00
0.02
0.04
0.06
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)
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
14
16
7/28/2019 Much as Much as Dis Tri Buci Ones
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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
i1i
xx
pp)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
7/28/2019 Much as Much as Dis Tri Buci Ones
13/75
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})
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
-1 0 1 2 3 4 5 6
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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
1ii
xx
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
7/28/2019 Much as Much as Dis Tri Buci Ones
15/75
Skewness:
No Closed Form
Kurtosis:
No Closed Form
Mode:
No Closed Form
PDF - CumulD({1,2,3,4},{.2,.3,.7,.8})
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
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
7/28/2019 Much as Much as Dis Tri Buci Ones
16/75
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
1i
ip)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:
=
i
N
1i
ipx
7/28/2019 Much as Much as Dis Tri Buci Ones
17/75
Variance:
Vp)x( i2
N
1i
i =
Skewness:
i3
N
1i
i23p)x(
V
1 =
Kurtosis:
i4
N
1i
i2p)x(
V
1 =
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})
0.00
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
7/28/2019 Much as Much as Dis Tri Buci Ones
18/75
Discrete Uniform
RISKDUniform({x})
Parameters:
{x} = {x1, x1, , xN} array of continuous parameters
Domain:
}x{x discrete
Mass and Cumulative Functions:
N
1)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:
=
N
1ii
xN
1
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19/75
Variance:
V)x(N
1 2N
1i
i =
Skewness:
3N
1i
i23)x(
NV
1 =
Kurtosis:
4N
1i
i2)x(
NV
1 =
Mode:
Not uniquely defined
PMF - DUniform({1,5,8,11,12})
0.00
0.05
0.10
0.15
0.20
0.25
0 2 4 6 810
12
14
CDF - DUniform({1,5,8,11,12})
0.0
0.2
0.4
0.6
0.8
1.0
0 2 4 6 810
12
14
7/28/2019 Much as Much as Dis Tri Buci Ones
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Error Function
RISKErf(h)
Parameters:
h continuous inverse scale parameter h > 0
Domain:
- x + continuous
Density and Cumulative Functions:
( )2hxeh
)x(f
=
( )hx2)x(F =
where is theError Function.
Mean:
0
Variance:
2h2
1
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Skewness:
0
Kurtosis:
3
Mode:
0
PDF - Erf(1)
0.0
0.1
0.2
0.3
0.4
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
0.2
0.3
0.4
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
7/28/2019 Much as Much as Dis Tri Buci Ones
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Erlang
RISKErlang(m,)
Parameters:
m integral shape parameter m > 0
continuous scale parameter > 0
Domain:
0 x < + continuous
Density and Cumulative Functions:
= x
1
ex
)!1m(
1)x(f
( )
( )
( )
=
=
=
1m
0i
ixx
!i
xe1
m
m)x(F
where is the Gamma Function and x is theIncomplete Gamma Function.
Mean:
m
Variance:
2m
7/28/2019 Much as Much as Dis Tri Buci Ones
23/75
Skewness:
m
2
Kurtosis:
m
63+
Mode:
( )1m
PDF - Erlang(2,1)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
-1 0 1 2 3 4 5 6 7
CDF - Erlang(2,1)
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
7/28/2019 Much as Much as Dis Tri Buci Ones
24/75
Exponential
RISKExpon()
Parameters:
continuous scale parameter > 0
Domain:
0 x < + continuous
Density and Cumulative Functions:
=
xe)x(f
= xe1)x(F
Mean:
Variance:
2
7/28/2019 Much as Much as Dis Tri Buci Ones
25/75
Skewness:
2
Kurtosis:
9
Mode:
0
PDF - Expon(1)
0.0
0.2
0.4
0.6
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
0.4
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
7/28/2019 Much as Much as Dis Tri Buci Ones
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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
1
b
1)x(f
)zexp(e
1)x(F
= where
( )
b
axz
Mean:
( ) b577.a1ba +
where (x) is the derivative of the Gamma Function.
Variance:
6
b22
7/28/2019 Much as Much as Dis Tri Buci Ones
27/75
Skewness:
1.139547
Kurtosis:
5.4
Mode:
a
PDF - ExtValue(0,1)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
-2
-1 0 1 2 3 4 5
CDF - ExtValue(0,1)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
-2
-1 0 1 2 3 4 5
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28/75
Gamma
RISKGamma(, )
Parameters:
continuous shape parameter > 0
continuous scale parameter > 0
Domain:
0 < x < + continuous
Density and Cumulative Functions:
( )
= x
1
ex1
)x(f
( )
( )
=
x)x(F
where is the Gamma Function and x is theIncomplete Gamma Function.
Mean:
Variance:
2
7/28/2019 Much as Much as Dis Tri Buci Ones
29/75
Skewness:
2
Kurtosis:
+
63
Mode:
( )1 if 1
Not Defined if < 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)
0.00
0.05
0.10
0.15
0.20
0.25
-2 0 2 4 6 8
10
12
7/28/2019 Much as Much as Dis Tri Buci Ones
30/75
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
xx
xxp)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
7/28/2019 Much as Much as Dis Tri Buci Ones
31/75
Skewness:
No Closed Form
Kurtosis:
No Closed Form
Mode:
No Closed Form
PDF - General(0,5,{1,2,3,4},{2,1,2,1})
0.00
0.05
0.10
0.15
0.20
0.25
0.30
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
7/28/2019 Much as Much as Dis Tri Buci Ones
32/75
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:
2p
p1
7/28/2019 Much as Much as Dis Tri Buci Ones
33/75
Skewness:
( )
p1
p2
Kurtosis:
( )2p1
p9
+
Mode:
0
PMF - Geomet(.5)
0.0
0.1
0.2
0.3
0.4
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
7/28/2019 Much as Much as Dis Tri Buci Ones
34/75
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
+
N
minmaximinx i
The {p} array has been normalized to give the histogram unit area.
Mean:
No Closed Form
Variance:
No Closed Form
7/28/2019 Much as Much as Dis Tri Buci Ones
35/75
Skewness:
No Closed Form
Kurtosis:
No Closed Form
Mode:
Not Uniquely Defined.
PDF - Histogrm(0,5,{6,5,3,4,5})
0.00
0.05
0.10
0.15
0.20
0.25
0.30
-1 0 1 2 3 4 5 6
CDF - Histogrm(0,5,{6,5,3,4,5})
0.0
0.2
0.4
0.6
0.8
1.0
-1 0 1 2 3 4 5 6
7/28/2019 Much as Much as Dis Tri Buci Ones
36/75
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:
=
n
M
xn
DM
x
D
)x(f
=
=
x
1i
n
M
xn
DM
x
D
)x(F
Mean:
MnD
7/28/2019 Much as Much as Dis Tri Buci Ones
37/75
Variance:
( )( )
( )
1M
nMDM
M
nD2
for M>1
Skewness:
( )( )
( ) )nM(DMnD
1M
2M
n2MD2M
for M>2
Mode:
(bimodal) xm and xm-1 if xm is integral
(unimodal) biggest integer less than xm otherwise
where( )( )
2M
1D1nxm
+
++
PMF - HyperGeo(6,5,10)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
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
7/28/2019 Much as Much as Dis Tri Buci Ones
38/75
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
+=
1minmax
1minx)x(F
+
+=
Mean:
2
maxmin+
Variance:
+
1
2
minmax
6
minmax
7/28/2019 Much as Much as Dis Tri Buci Ones
39/75
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
7/28/2019 Much as Much as Dis Tri Buci Ones
40/75
Inverse Gaussian
RISKInvGauss(, )
Parameters:
continuous parameter > 0
continous parameter > 0
Domain:
x > 0 continuous
Density and Cumulative Functions:
( )
=
x2
x
3
2
2
ex2
)x(f
+
+
= 1
x
x
e1x
x
)x(F 2
where is theError Function.
Mean:
Variance:
3
7/28/2019 Much as Much as Dis Tri Buci Ones
41/75
Skewness:
3
Kurtosis:
+153
Mode:
+
2
3
4
91
2
2
PDF - InvGauss(1,2)
0.0
0.2
0.4
0.6
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
CDF - InvGauss(1,2)
0.0
0.1
0.2
0.3
0.4
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
7/28/2019 Much as Much as Dis Tri Buci Ones
42/75
Logistic
RISKLogistic(, )
Parameters:
continuous location parameter
continuous scale parameter > 0
Domain:
- x + continuous
Density and Cumulative Functions:
=4
x
2
1hsec
)x(f
2
2
x
2
1tanh1
)x(F
+
=
where sech is theHyperbolic Secant Function and tanh is theHyperbolic Tangent Function.
Mean:
Variance:
3
22
7/28/2019 Much as Much as Dis Tri Buci Ones
43/75
Skewness:
0
Kurtosis:
4.2
Mode:
PDF - Logistic(0,1)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
-5
-4
-3
-2
-1 0 1 2 3 4 5
CDF - Logistic(0,1)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
-5
-4
-3
-2
-1 0 1 2 3 4 5
7/28/2019 Much as Much as Dis Tri Buci Ones
44/75
Log-Logistic
RISKLogLogistic(, , )
Parameters:
continuous location parameter
continous scale parameter > 0
continuous shape parameter > 0
Definitions:
Domain:
x + continuous
Density and Cumulative Functions:
( )21
t1
t)x(f
+
=
+
=
t
11
1)x(F with
xt
Mean:
( ) + csc for > 1
7/28/2019 Much as Much as Dis Tri Buci Ones
45/75
Variance:
( ) ( )[ ] 22 csc2csc2 for > 2
Skewness:
( ) ( ) ( ) ( )
( ) ( )[ ] 23
2
32
csc2csc2
csc2csc2csc63csc3
+ for > 3
Mode:
++
1
11 for > 1
0 for 1
PDF - LogLogistic(0,1,5)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
-0
.5
0.
0
0.
5
1.
0
1.
5
2.
0
2.
5
3.
0
CDF - LogLogistic(0,1,5)
0.0
0.1
0.2
0.3
0.4
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
7/28/2019 Much as Much as Dis Tri Buci Ones
46/75
Lognormal (Format 1)
RISKLognorm(, )
Parameters:
continuous parameter > 0
continous parameter > 0
Domain:
0 x + continuous
Density and Cumulative Functions:
2xln
2
1
e2x
1)x(f
=
=
xln)x(F
with
+
22
2
ln and
+
2
1ln
where is theError Function.
Mean:
Variance:
2
7/28/2019 Much as Much as Dis Tri Buci Ones
47/75
Skewness:
+
3
3
Kurtosis:
332234 ++ with
+ 1
Mode:
( ) 23224
+
PDF - Lognorm(1,1)
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
CDF - Lognorm(1,1)
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
7/28/2019 Much as Much as Dis Tri Buci Ones
48/75
Lognormal (Format 2)
RISKLognorm2(, )
Parameters:
continuous parameter
continous parameter > 0
Domain:
0 x + continuous
Density and Cumulative Functions:
2xln
2
1
e2x
1)x(f
=
=
xln)x(F
where is theError Function.
Mean:
2
2
e
+
Variance:
( )1e2 with2
e
7/28/2019 Much as Much as Dis Tri Buci Ones
49/75
Skewness:
( ) 12 + with2
e
Kurtosis:
332 234 ++ with2
e
Mode:
2e
PDF - Lognorm2(0,1)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
-2 0 2 4 6 8
10
12
CDF - Lognorm2(0,1)
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
7/28/2019 Much as Much as Dis Tri Buci Ones
50/75
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 theBinomial Coefficient.
Mean:
( )
p
p1s
Variance:
( )2
p
p1s
7/28/2019 Much as Much as Dis Tri Buci Ones
51/75
Skewness:
( )p1s
p2
Kurtosis:
( )p1s
p
s
63
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)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
-1 0 1 2 3 4 5 6 7 8 9
CDF - NegBin(3,.6)
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 8 9
7/28/2019 Much as Much as Dis Tri Buci Ones
52/75
Normal
RISKNormal(, )
Parameters:
continuous location parameter
continuous scale parameter > 0
Domain:
- x + continuous
Density and Cumulative Functions:
2x
2
1
e2
1)x(f
=
=
x)x(F
where is theError Function.
Mean:
Variance:
2
7/28/2019 Much as Much as Dis Tri Buci Ones
53/75
Skewness:
0
Kurtosis:
3
Mode:
PDF - Normal(0,1)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
-3
-2
-1 0 1 2 3
CDF - Normal(0,1)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
-3
-2
-1 0 1 2 3
7/28/2019 Much as Much as Dis Tri Buci Ones
54/75
Pareto (First Kind)
Pareto(, a)
Parameters:
continuous shape parameter > 0
a continuous scale parameter a > 0
Domain:
a x + continuous
Density and Cumulative Functions:
1x
a)x(f
+
=
=
x
a1)x(F
Mean:
1
a
for > 1
Variance:
( ) ( )21a2
2
for > 2
7/28/2019 Much as Much as Dis Tri Buci Ones
55/75
Skewness:
+ 2
3
12 for > 3
Kurtosis:
( )
( )( )43
2323 2
++for > 4
Mode:
a
PDF - Pareto(2,1)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0 1 2 3 4 5 6 7 8 910
11
CDF - Pareto(2,1)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 1 2 3 4 5 6 7 8 910
11
7/28/2019 Much as Much as Dis Tri Buci Ones
56/75
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
bx
qb)x(f
++=
( )q
q
bx
b1)x(F
+=
Mean:
1q
b
for q > 1
Variance:
( ) ( )2q1q
qb
2
2
for q > 2
7/28/2019 Much as Much as Dis Tri Buci Ones
57/75
Skewness:
+
3q
1q
q
2q2 for q > 3
Mode:
0
PDF - Pareto2(3,3)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-2 0 2 4 6 8
10
12
CDF - Pareto2(3,3)
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
7/28/2019 Much as Much as Dis Tri Buci Ones
58/75
Pearson Type V
RISKPearson5(, )
Parameters:
continuous shape parameter > 0
continous scale parameter > 0
Domain:
0 x < + continuous
Density and Cumulative Functions:
( ) ( ) 1
x
x
e1)x(f
+
=
F(x) Has No Closed Form
Mean:
1
for > 1
Variance:
( ) ( )212
2
for > 2
7/28/2019 Much as Much as Dis Tri Buci Ones
59/75
Skewness:
3
24
for > 3
Kurtosis:
( )( )( )( )43
253
+ for > 4
Mode:
1+
PDF - Pearson5(3,1)
0.0
0.5
1.0
1.5
2.0
2.5
-0
.5
0.
0
0.
5
1.
0
1.
5
2.
0
2.
5
CDF - Pearson5(3,1)
0.0
0.1
0.2
0.3
0.4
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
7/28/2019 Much as Much as Dis Tri Buci Ones
60/75
Pearson Type VI
RISKPearson6(1, 2, )
Parameters:
1 continuous shape parameter 1> 0
2 continuous shape parameter 2> 0
continuous scale parameter > 0
Domain:
0 x < + continuous
Density and Cumulative Functions:
( )
( )21
1
x1
x
,B
1)x(f
1
21+
+
=
F(x) Has No Closed Form.
where B is theBeta Function.
Mean:
12
1
for2 > 1
Variance:
( )
( ) ( )21
1
22
2
2112
+for2 > 2
7/28/2019 Much as Much as Dis Tri Buci Ones
61/75
Skewness:
( )
+
+
3
12
1
22
2
21
211
2 for2 > 3
Kurtosis:
( )
( )( )
( )
( )( )
++
+
5
1
12
43
232
211
22
22
2 for2 > 4
Mode:
( )1
1
2
1
+
for1 > 1
0 otherwise
PDF - Pearson6(3,3,1)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
-1 0 1 2 3 4 5 6 7 8 9
CDF - Pearson6(3,3,1)
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 8 9
7/28/2019 Much as Much as Dis Tri Buci Ones
62/75
Pert (Beta)
RISKPert(min, m.likely, max)
Definitions:
6
maxlikely.m4min ++
minmax
min61
minmax
max62
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
+
=
( )( )21z
21
21z ,I),(B
,B)x(F
= with
minmax
minxz
where B is theBeta Function and Bz is theIncomplete Beta Function.
Mean:
6
maxlikely.m4min ++
7/28/2019 Much as Much as Dis Tri Buci Ones
63/75
Variance:
( )( )7
maxmin
Skewness:
( )( )+
maxmin
7
4
2maxmin
Kurtosis:
( ) ( ) ( )( )( )32
6213212121
21212
2121++++
+++++
Mode:
m.likely
PDF - Pert(0,1,3)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
-0
.5
0.
0
0.
5
1.
0
1.
5
2.
0
2.
5
3.
0
3.
5
CDF - Pert(0,1,3)
0.0
0.2
0.4
0.6
0.8
1.0
-0
.5
0.
0
0.
5
1.
0
1.
5
2.
0
2.
5
3.
0
3.
5
7/28/2019 Much as Much as Dis Tri Buci Ones
64/75
Poisson
RISKPoisson()
Parameters:
mean number of successes continuous > 0
Domain:
0 x < + discrete
Mass and Cumulative Functions:
!x
e)x(f
x =
=
=
x
0n
n
!ne)x(F
Mean:
Variance:
7/28/2019 Much as Much as Dis Tri Buci Ones
65/75
Skewness:
1
Kurtosis:
+
13
Mode:
(bimodal) and +1 (bimodal) if is an integer
(unimodal) largest integer less than otherwise
PMF - Poisson(3)
0.00
0.05
0.10
0.15
0.20
0.25
-1 0 1 2 3 4 5 6 7 8 9
CDF - Poisson(3)
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 8 9
7/28/2019 Much as Much as Dis Tri Buci Ones
66/75
Rayleigh
RISKRayleigh(b)
Parameters:
b continuous scale parameter b> 0
Domain:
0 x < + continuous
Density and Cumulative Functions:
2
b
x
2
1
2e
b
x)x(f
=
2
b
x
2
1
e1)x(F
=
Mean:
2b
Variance:
2
2b2
7/28/2019 Much as Much as Dis Tri Buci Ones
67/75
Skewness:
( )
( )6311.0
4
32
23
Kurtosis:
( )2451.3
4
332
2
2
Mode:
b
PDF - Rayleigh(1)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
-0
.5
0.
0
0.
5
1.
0
1.
5
2.
0
2.
5
3.
0
3.
5
CDF - Rayleigh(1)
0.0
0.1
0.2
0.3
0.4
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
7/28/2019 Much as Much as Dis Tri Buci Ones
68/75
Students t
RISKStudent()
Parameters:
the degrees of freedom integer > 0
Domain:
- x + continuous
Density and Cumulative Functions:
2
1
2x
2
2
1
1)x(f
+
+
+
=
+=
2,
2
1I1
2
1)x(F s with 2
2
x
xs
+
where is the Gamma Function and Ix is theIncomplete Beta Function.
Mean:
0 for > 1*
*even though the mean is not well defined for = 1, the distribution is still symmetrical about 0.
Variance:
2
for > 2
7/28/2019 Much as Much as Dis Tri Buci Ones
69/75
Skewness:
0 for > 3*
*even though the skewness is not well defined for 3, the distribution is still symmetric about 0.
Kurtosis:
4
23 for > 4
Mode:
0
PDF - Student(3)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
-5
-4
-3
-2
-1 0 1 2 3 4 5
CDF - Student(3)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
-5
-4
-3
-2
-1 0 1 2 3 4 5
7/28/2019 Much as Much as Dis Tri Buci Ones
70/75
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(max
xmax2)x(f
= m.likely x max
( )
( )( )minmaxminlikely.mminx
)x(F2
= min x m.likely
( )
( )( )minmaxlikely.mmax
xmax1)x(F
2
= m.likely x max
Mean:
3
maxlikely.mmax ++
7/28/2019 Much as Much as Dis Tri Buci Ones
71/75
Variance:
( )( ) ( )( ) ( )( )
18
minmaxminlikely.mlikely.mmaxmaxlikely.mmax 222 ++
Skewness:
( ) 2322
3f
9ff
5
22
+
where
( )1
minmax
minlikely.m2f
Kurtosis:
2.4
Mode:
m.likely
PDF - Triang(0,3,5)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
-1 0 1 2 3 4 5 6
CDF - Triang(0,3,5)
0.0
0.2
0.4
0.6
0.8
1.0
-1 0 1 2 3 4 5 6
7/28/2019 Much as Much as Dis Tri Buci Ones
72/75
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
=
minmax
minx)x(F
=
Mean:
2
minmax
Variance:
( )12
minmax2
7/28/2019 Much as Much as Dis Tri Buci Ones
73/75
Skewness:
0
Kurtosis:
1.8
Mode:
Not uniquely defined
CDF - Uniform(0,1)
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
PDF - Uniform(0,1)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
-0
.2
0.
0
0.
2
0.
4
0.
6
0.
8
1.
0
1.
2
7/28/2019 Much as Much as Dis Tri Buci Ones
74/75
Weibull
RISKWeibull(, )
Parameters:
continuous shape parameter > 0
continuous scale parameter > 0
Domain:
0 x < + continuous
Density and Cumulative Functions:
( )
= x
1
ex
)x(f
( )= xe1)x(F
Mean:
+
11b
where is the Gamma Function.
Variance:
+
+
11
21
22
where is the Gamma Function.
7/28/2019 Much as Much as Dis Tri Buci Ones
75/75
Skewness:
23
2
3
1121
112
11
213
31
+
+
++
+
++
+
where is the Gamma Function.
Mode:
11
1 for >1
0 for 1
PDF - Weibull(2,1)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
-0
.5
0.
0
0.
5
1.
0
1.
5
2.
0
2.
5
CDF - Weibull(2,1)
0.0
0.1
0.2
0.3
0.4
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