PROBABILITY DISTRIBUTION FUNCTIONS

Reliability/Survival Software Engines

Probability Distribution Function

Input

Automatic Output and Charts

PDF

PDF Name

PDF Type

Alternative Name

PDF Bank

Number of PDF Parameters

Random or Time Variables

Probability Distribution Function

Reliability / Survival

Unreliability

Failure / Hazard Rate

Cumulative Hazard Function

Conditional Reliability / Survival

Conditional Unreliability

Probability of Failure During Mission

Worksheet

Parametric Model

Version or Model

Name/Category/Subfamilies

Workbook

Age at Start of Mission

Mission Time

f(T) and f(T+t)1

R(T) and R(T+t)2

Q(T) and Q(T+t)2

λ(T) and λ(T+t)

H(T) and H(T+t)

R(T,t)

Q(T,t)

Q(T+t)-Q(T)

1

8020.D

α%-(1-α%) Rule

Discrete

Includes 80%-20% Rule

Discrete

1

T

t

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2

AMO

Amoroso

Generalized Distribution Hierarchy

Distribution Subfamilies: Stacy (Generalized Gamma/Inverse Gamma), Stretched Exponential; Gamma, Wilson-Hilferty, Pearson Type III, Exponential, Shifted Exponential, Half-Normal, Nakagami, Chi-Square, Scaled-Chi-Square, Chi, Scaled-Chi, Rayleigh, Maxwell; Pearson Type V, Inverse Gamma, Inverse Exponential, Lévy, Scaled Inverse Chi-Square, Inverse-Chi Square, Scaled Inverse-Chi, Inverse-Chi, Inverse Rayleigh; Generalized Fisher-Tippett, Fisher-Tippett (Generalized Extreme Value), Weibull, Generalized Weibull, Fréchet, Generalized Fréchet4

PDF-A

4

T

t

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3

BETA

Beta

Euler Beta of the First Kind; Pearson Type I

PDF-B

2

T

t

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4

BETA4

Beta

PDF-B

4

T

t

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5

BETAP

Beta Prime

Pearson Type VI; Inverted Beta; Beta Distribution of the 2nd Kind

PDF-B

4

T

t

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6

BIN.D

Binomial

Discrete

Discrete

2

T

t

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7

BRAD

Bradford

Bradford Law of Scattering

PDF-B

3

T

t

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8

BS

Birnbaum-Saunders

Fatigue Life

PDF-B

3

T

t

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9

BUR1

Burr

XVII

Singh-Maddala; Generalized Log-Logistic

PDF-B

2

T

t

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10

BUR2

Burr

XVII

Transformed Pareto

PDF-B

3

T

t

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11

BUR3

Burr

XVII

PDF-B

3

T

t

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12

BUR4

Burr

XVII

PDF-B

4

T

t

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13

CAU

Cauchy

PDF-C

2

T

t

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14

CAUT

Doubly Truncated Cauchy

PDF-C

4

T

t

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15

CHI

Chi

PDF-C

1

T

t

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16

CHI3

Chi

PDF-C

3

T

t

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17

CHIS

Chi-Square

Standard

PDF-C

1

T

t

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18

CHIS3

Chi-Square

PDF-C

3

T

t

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19

DAG

Dagum

PDF-D

3

T

t

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20

DAG1

Dagum

1

PDF-D

3

T

t

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21

DAGIB

Dagum

Inverse Burr; Kappa

PDF-D

4

T

t

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22

EV1MAX

Extreme Value for Maxima

1

Gumbell

PDF-E

2

T

t

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23

EV1MIN

Extreme Value for Minima

1

PDF-E

2

T

t

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24

EV2MAX

Extreme Value for Maxima

2

Fréchet; Reversed Weibull

PDF-E

3

T

t

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25

EV2MIN

Extreme Value for Minima

2

PDF-E

3

T

t

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26

EV3MAX

Extreme Value for Maxima

3

PDF-E

3

T

t

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27

EV3MIN

Extreme Value for Minima

3

Weibull

PDF-E

3

T

t

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28

EXP

Exponential

PDF-E

2

T

t

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29

EXPL

Exponential-Logarithmic

PDF-E

2

T

t

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30

F

F-Distribution

Standard

Fisher-Snedecor

PDF-F

2

T

t

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31

F4

F-Distribution

Fisher-Snedecor

PDF-F

4

T

t

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32

FP

Feller-Pareto

Generalized Distribution Hierarchy

Distribution Subfamilies: Pareto; Transformed Beta (Burr, Log-Logistic, Paralogistic); Generalized Pareto (Scaled-F, Inverse Pareto); Inverse Burr (Log-Logistic, Inverse Pareto, Inverse Paralogistic)5

PDF-F

5

T

t

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33

FT

Fisher-Tippett

Generalized Extreme Value (GEV); von Mises-Jenkinson; von Mises Extreme Value4

PDF-F

3

T

t

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34

GAM

Gamma

Pearson Type III

PDF-G

3

T

t

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35

GBUR

Generalized Burr

PDF-G

5

T

t

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36

GEO.D

Geometric

Discrete

Discrete

1

T

t

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37

GEV

Generalized Extreme Value

Generalized Distribution Hierarchy

Fisher-Tippett Distribution; Distribution Subfamilies4,6: Gumbell, Fréchet, and Reversed Weibull (Extreme Value Distributions Types I, II, and III)6

PDF-G

3

T

t

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38

GFRE

Generalized Fréchet

PDF-G

4

T

t

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39

GFT

Generalized Fisher-Tippett

PDF-G

4

T

t

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40

GGAM

Generalized Gamma

PDF-G

4

T

t

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41

GLOG

Generalized Logistic

PDF-G

3

T

t

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42

GLOG7

Generalized Logistic

Generalized Distribution Hierarchy

Generalized Logistic Distribution Subfamilies7: Ojo-Olapade (6 Parameters); Wu et al. (5 Parameters); George-Ojo (4 Parameters); Extended Types I, II (4 Parameters); Balakrishnan-Leung Types I, II, III (3 Parameters); Berkson (2 Parameters); Located/Scaled (1 Parameter); Standard (0 Parameters)

PDF-G

7

T

t

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42

GMAK

Generalized Makeham

Kosznik-Biernacka

PDF-G

3

T

t

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43

GN1

Generalized Normal

1

PDF-G

3

T

t

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44

GN2

Generalized Normal

2

PDF-G

3

T

t

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45

GOM

Gompertz

PDF-G

2

T

t

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46

GP

Generalized Pareto

PDF-G

3

T

t

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47

GPB2

Generalized Pareto

Beta of the Second Kind

PDF-G

3

T

t

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48

GW

Generalized Weibull

PDF-G

4

T

t

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49

GY.D

Generalized Yule

Discrete

Discrete

2

T

t

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50

HGEO.D

Hpypergeometric

Discrete

Discrete

3

T

t

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51

HS

Hyperbolic Secant

PDF-H

2

T

t

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52

HYPER

Hyperexponential

PDF-H

10

T

t

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53

HYPO

Hypoexponential

PDF-H

5

T

t

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54

IGAM

Inverse Gamma

Vinci

PDF-I

2

T

t

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55

IGAMPV

Inverse Gamma

Pearson Type V

PDF-I

3

T

t

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56

IGAUS

Inverse Gaussian

Inverse Normal - Schrödinger-Wald

PDF-I

2

T

t

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57

IPL

Inverse Paralogistic

PDF-I

2

T

t

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58

JSU

Johnson Unbounded

PDF-J

4

T

t

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59

JSB

Johnson Bounded

PDF-J

4

T

t

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60

KUM

Kumaraswamy Doubly Bounded

PDF-K

2

T

t

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61

KUM4

Kumaraswamy Doubly Bounded

PDF-K

4

T

t

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62

L

Logarithmic

PDF-L

2

T

t

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63

LAP

Laplace

PDF-L

2

T

t

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64

LEV

Lévy

Van Der Waals Profile

PDF-L

2

T

t

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65

LEXP

Linear Exponential

PDF-L

2

T

t

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66

LLAP

Log-Laplace

Double Pareto

PDF-L

3

T

t

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67

LLOG

Log-Logistic

Fisk

PDF-L

3

T

t

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68

LN

LogNormal

PDF-L

2

T

t

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69

LOG

Logistic

PDF-L

2

T

t

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70

LOM

Lomax

Pareto-Lomax

PDF-L

2

T

t

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71

LS.D

Log-Series

Discrete

Logarithmic

Discrete

1

T

t

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72

MAK

Makeham

PDF-M

3

T

t

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73

MIX3

Multiple Failure Modes

Mixture

Mixed Population Model (1-3 Subpopulations)

PDF-M

3

T

t

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74

MIX6

Multiple Failure Modes

Mixture

Mixed Population Model (1-6 Subpopulations)

PDF-M

5

T

t

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75

NAK

Nakagami-m

Fading Model

PDF-N

2

T

t

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76

NAKGN

Nakagami

Generalized Normal

PDF-N

3

T

t

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77

N

Normal

Gaussian; Bell Curve; Pearson Type V

PDF-N

2

T

t

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78

NT

Doubly Truncated Normal

PDF-N

4

T

t

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79

NTL

Left Truncated Normal

PDF-N

3

T

t

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80

NTR

Right Truncated Normal

PDF-N

3

T

t

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81

NBIN.D

Negative Binomial

Discrete

Pascal

Discrete

2

T

t

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82

PIII

Pearson

III

PDF-P

3

T

t

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83

P1

Pareto 1

PDF-P

2

T

t

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84

P2

Pareto 2

PDF-P

3

T

t

●

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85

P3

Pareto 3

PDF-P

3

T

t

●

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86

P4

Pareto 4

PDF-P

4

T

t

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87

PB

Bounded Pareto

PDF-P

3

T

t

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88

PERT

PERT

Standard

PDF-P

3

T

t

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89

PERTV

PERT

Modified

Vose

PDF-P

4

T

t

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90

PL

Paralogistic

PDF-P

2

T

t

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91

PN

Power Normal

PDF-P

3

T

t

●

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92

POI.D

Poisson

Discrete

Discrete

1

T

t

●

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93

RAY

Rayleigh

PDF-R

1

T

t

●

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94

RBT1

Reliability Bathtub

1

Dhillon

PDF-R

2

T

t

●

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95

RBT2

Reliability Bathtub

2

Hjorth

PDF-R

3

T

t

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96

RBT3

Reliability Bathtub

3

Govil-Aggarwal

PDF-R

6

T

t

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97

RBT4

Reliability Bathtub

4

Dhillon

PDF-R

5

T

t

●

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98

SGEO.D

Shifted Geometric

Discrete

Discrete

1

T

t

●

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99

SGOMP

Shifted Gompertz

PDF-S

2

T

t

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100

SIL

Siler

Siler Competing Hazards Model

PDF-S

5

T

t

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101

SLLOG

Shifted Log-Logistic

Generalized Log-Logistic

PDF-S

3

T

t

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102

STACY

Stacy

Distribution Subfamilies: Generalized Gamma; Generalized Inverse Gamma4

PDF-S

3

T

t

●

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103

SUNI.D

Step Uniform

Discrete

Discrete

3

T

t

●

●

●

●

●

●

●

●

104

T

Student t

Standard

Gosset; Pearson Type IV

PDF-T

1

T

t

●

●

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●

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●

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105

T3

Student t

PDF-T

3

T

t

●

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106

TB

Transformed Beta

PDF-T

4

T

t

●

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107

TRI

Triangular

PDF-T

3

T

t

●

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108

UNI

Uniform

PDF-U

2

T

t

●

●

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●

●

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●

109

UNI.D

Uniform

Discrete

Discrete

2

T

t

●

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●

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●

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●

●

110

UQ

U-Quadratic

PDF-U

2

T

t

●

●

●

●

●

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●

●

111

WEI

Weibull

Extreme Value Type III; Fisher-Tippett Type III; Gumbel Type III

PDF-W

3

T

t

●

●

●

●

●

●

●

●

112

WEIL

Weibull-Logarithmic

Ciumara-Preda

PDF-W

3

T

t

●

●

●

●

●

●

●

●

113

WH

Wilson-Hilferty

PDF-W

2

T

t

●

●

●

●

●

●

●

●

114

WO

Wearout Life

Zacks

PDF-W

3

T

t

●

●

●

●

●

●

●

●

115

YS.D

Yule-Simon

Discrete

Discrete

1

T

t

●

●

●

●

●

●

●

●

116

ZIPF.D

Zipf's Law

Discrete

Discrete

2

T

t

●

●

●

●

●

●

●

●

117

ZM.D

Zipf-Mandelbrot Law

Discrete

Discrete

3

T

t

●

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Notes

1 Probability Density Function (PDF).

2 Cumulative Distribution Function (CDF) (Upper/Lower). Reliability=Upper CDF; Unreliability=Lower CDF.

4 PDF Banks: Continuous PDFs, organized alphabetically in workbooks PDF-A through PDF-Z; Discrete PDFs in workbook PDF-Discrete.

4 Gavin E. Crooks, The Amoroso Distribution. Lawrence Berkeley National Laboratory, U.S. Department of Energy, Tech. Note 003v4 (2010-01-14), http://threeplusone.com/pubs/technote/CrooksTechNote003.pdf. See also arXiv:1005.3274v1 [math.ST] 18 May 2010, http://arxiv.org/PS_cache/arxiv/pdf/1005/1005.3274v1.pdf.

5 S. Tahmasebi and J. Behboodian, Shannon Entropy for the Feller-Paret (FP) Family and Order Statistics of FP Subfamilies. Applied Mathematical Sciences, Vol. 4, 2010, no. 10, 495-505, http://www.m-hikari.com/ams/ams-2010/ams-9-12-2010/tahmasebiAMS9-12-2010.pdf.

6 Generalized Extreme Value Distribution (GEV). Wikipedia, 21-Apr-2010, http://en.wikipedia.org/wiki/Generalized_extreme_value_distribution.

PDF Bank

PDF Type

Number of PDFs

Automatic Output Functions/Charts

A-Z

Continuous

104

1,352

Discrete

Discrete

14

182

All

Continuous and Discrete

118

1,534

Macroknow probability distributions are used in many fields: engineering, finance, commerce, industry, military, medical, information science, research, etc. Industries where statistical techniques are extremely useful include manufacturing (automotive, aerospace, electronics), chemical process plants, energy (power generation, nuclear, oil, gas), transportation (airlines, railways), medical (pharmaceutical, hospitals), finance (banking, insurance), security, and defense. Applications include: reliability engineering, event tree analysis, fault tree analysis, attack tree analysis, survival data analysis, expert systems, insurance, actuarial science, and demographics. Depending on the probability distribution, input variables can include, one or more location, scale, and shape parameters. The input random variables for the probability distributions are the age T of a physical or biological unit and a mission time or life period t. The physical unit can be a system, a device, a component, or a part (mechanical, electrical, or electronic). The biological unit can be a human, a patient, an animal, or a living cell. The output variables include: the probability density function (PDF), the upper cumulative distribution function (reliability or survival function), the lower cumulative distribution function (CDF) (unreliability or death function), the hazard rate (failure rate or force of mortality), the cumulative hazard function, the conditional reliability or survival function, and the conditional unreliability or death function. In engineering, the conditional reliability is the probability that the physical unit will continue to operate during the mission time t, given that it has already operated for a time T. In medical or actuarial applications, the conditional survival is the probability that a person aged T, survives an additional time period t. All output variables are graphed automatically.

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