PROBABILITY DISTRIBUTION FUNCTIONS | ||||||||||||||||

Reliability/Survival Software Engines | ||||||||||||||||

Probability Distribution Function | Input | Automatic Output and Charts | ||||||||||||||

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 | ● | ● | ● | ● | ● | ● | ● | ● |

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 Fréchet^{4} |
PDF-A | 4 | T | t | ● | ● | ● | ● | ● | ● | ● | ● |

3 | BETA | Beta | Euler Beta of the First Kind; Pearson Type I | PDF-B | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

4 | BETA4 | Beta | PDF-B | 4 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

5 | BETAP | Beta Prime | Pearson Type VI; Inverted Beta; Beta Distribution of the 2nd Kind | PDF-B | 4 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

6 | BIN.D | Binomial | Discrete | Discrete | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

7 | BRAD | Bradford | Bradford Law of Scattering | PDF-B | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

8 | BS | Birnbaum-Saunders | Fatigue Life | PDF-B | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

9 | BUR1 | Burr | XVII | Singh-Maddala; Generalized Log-Logistic | PDF-B | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● |

10 | BUR2 | Burr | XVII | Transformed Pareto | PDF-B | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● |

11 | BUR3 | Burr | XVII | PDF-B | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

12 | BUR4 | Burr | XVII | PDF-B | 4 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

13 | CAU | Cauchy | PDF-C | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

14 | CAUT | Doubly Truncated Cauchy | PDF-C | 4 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

15 | CHI | Chi | PDF-C | 1 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

16 | CHI3 | Chi | PDF-C | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

17 | CHIS | Chi-Square | Standard | PDF-C | 1 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

18 | CHIS3 | Chi-Square | PDF-C | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

19 | DAG | Dagum | PDF-D | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

20 | DAG1 | Dagum | 1 | PDF-D | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

21 | DAGIB | Dagum | Inverse Burr; Kappa | PDF-D | 4 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

22 | EV1MAX | Extreme Value for Maxima | 1 | Gumbell | PDF-E | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● |

23 | EV1MIN | Extreme Value for Minima | 1 | PDF-E | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

24 | EV2MAX | Extreme Value for Maxima | 2 | Fréchet; Reversed Weibull | PDF-E | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● |

25 | EV2MIN | Extreme Value for Minima | 2 | PDF-E | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

26 | EV3MAX | Extreme Value for Maxima | 3 | PDF-E | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

27 | EV3MIN | Extreme Value for Minima | 3 | Weibull | PDF-E | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● |

28 | EXP | Exponential | PDF-E | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

29 | EXPL | Exponential-Logarithmic | PDF-E | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

30 | F | F-Distribution | Standard | Fisher-Snedecor | PDF-F | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● |

31 | F4 | F-Distribution | Fisher-Snedecor | PDF-F | 4 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

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 | ● | ● | ● | ● | ● | ● | ● | ● |

33 | FT | Fisher-Tippett | Generalized Extreme Value (GEV); von Mises-Jenkinson; von Mises
Extreme Value^{4} |
PDF-F | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

34 | GAM | Gamma | Pearson Type III | PDF-G | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

35 | GBUR | Generalized Burr | PDF-G | 5 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

36 | GEO.D | Geometric | Discrete | Discrete | 1 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

37 | GEV | Generalized Extreme Value | Generalized Distribution Hierarchy | Fisher-Tippett Distribution; Distribution Subfamilies^{4,6}: Gumbell,
Fréchet, and Reversed Weibull (Extreme Value Distributions Types I, II, and
III)^{6} |
PDF-G | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● |

38 | GFRE | Generalized Fréchet | PDF-G | 4 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

39 | GFT | Generalized Fisher-Tippett | PDF-G | 4 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

40 | GGAM | Generalized Gamma | PDF-G | 4 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

41 | GLOG | Generalized Logistic | PDF-G | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

42 | GLOG7 | Generalized Logistic | Generalized Distribution Hierarchy | Generalized Logistic Distribution Subfamilies^{7}: 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 | ● | ● | ● | ● | ● | ● | ● | ● |

42 | GMAK | Generalized Makeham | Kosznik-Biernacka | PDF-G | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

43 | GN1 | Generalized Normal | 1 | PDF-G | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

44 | GN2 | Generalized Normal | 2 | PDF-G | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

45 | GOM | Gompertz | PDF-G | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

46 | GP | Generalized Pareto | PDF-G | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

47 | GPB2 | Generalized Pareto | Beta of the Second Kind | PDF-G | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

48 | GW | Generalized Weibull | PDF-G | 4 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

49 | GY.D | Generalized Yule | Discrete | Discrete | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

50 | HGEO.D | Hpypergeometric | Discrete | Discrete | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

51 | HS | Hyperbolic Secant | PDF-H | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

52 | HYPER | Hyperexponential | PDF-H | 10 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

53 | HYPO | Hypoexponential | PDF-H | 5 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

54 | IGAM | Inverse Gamma | Vinci | PDF-I | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

55 | IGAMPV | Inverse Gamma | Pearson Type V | PDF-I | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

56 | IGAUS | Inverse Gaussian | Inverse Normal - Schrödinger-Wald | PDF-I | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

57 | IPL | Inverse Paralogistic | PDF-I | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

58 | JSU | Johnson Unbounded | PDF-J | 4 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

59 | JSB | Johnson Bounded | PDF-J | 4 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

60 | KUM | Kumaraswamy Doubly Bounded | PDF-K | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

61 | KUM4 | Kumaraswamy Doubly Bounded | PDF-K | 4 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

62 | L | Logarithmic | PDF-L | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

63 | LAP | Laplace | PDF-L | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

64 | LEV | Lévy | Van Der Waals Profile | PDF-L | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

65 | LEXP | Linear Exponential | PDF-L | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

66 | LLAP | Log-Laplace | Double Pareto | PDF-L | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

67 | LLOG | Log-Logistic | Fisk | PDF-L | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

68 | LN | LogNormal | PDF-L | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

69 | LOG | Logistic | PDF-L | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

70 | LOM | Lomax | Pareto-Lomax | PDF-L | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

71 | LS.D | Log-Series | Discrete | Logarithmic | Discrete | 1 | T | t | ● | ● | ● | ● | ● | ● | ● | ● |

72 | MAK | Makeham | PDF-M | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

73 | MIX3 | Multiple Failure Modes | Mixture | Mixed Population Model (1-3 Subpopulations) | PDF-M | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● |

74 | MIX6 | Multiple Failure Modes | Mixture | Mixed Population Model (1-6 Subpopulations) | PDF-M | 5 | T | t | ● | ● | ● | ● | ● | ● | ● | ● |

75 | NAK | Nakagami-m | Fading Model | PDF-N | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

76 | NAKGN | Nakagami | Generalized Normal | PDF-N | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

77 | N | Normal | Gaussian; Bell Curve; Pearson Type V | PDF-N | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

78 | NT | Doubly Truncated Normal | PDF-N | 4 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

79 | NTL | Left Truncated Normal | PDF-N | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

80 | NTR | Right Truncated Normal | PDF-N | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

81 | NBIN.D | Negative Binomial | Discrete | Pascal | Discrete | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● |

82 | PIII | Pearson | III | PDF-P | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

83 | P1 | Pareto 1 | PDF-P | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

84 | P2 | Pareto 2 | PDF-P | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

85 | P3 | Pareto 3 | PDF-P | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

86 | P4 | Pareto 4 | PDF-P | 4 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

87 | PB | Bounded Pareto | PDF-P | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

88 | PERT | PERT | Standard | PDF-P | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

89 | PERTV | PERT | Modified | Vose | PDF-P | 4 | T | t | ● | ● | ● | ● | ● | ● | ● | ● |

90 | PL | Paralogistic | PDF-P | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

91 | PN | Power Normal | PDF-P | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

92 | POI.D | Poisson | Discrete | Discrete | 1 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

93 | RAY | Rayleigh | PDF-R | 1 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

94 | RBT1 | Reliability Bathtub | 1 | Dhillon | PDF-R | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● |

95 | RBT2 | Reliability Bathtub | 2 | Hjorth | PDF-R | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● |

96 | RBT3 | Reliability Bathtub | 3 | Govil-Aggarwal | PDF-R | 6 | T | t | ● | ● | ● | ● | ● | ● | ● | ● |

97 | RBT4 | Reliability Bathtub | 4 | Dhillon | PDF-R | 5 | T | t | ● | ● | ● | ● | ● | ● | ● | ● |

98 | SGEO.D | Shifted Geometric | Discrete | Discrete | 1 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

99 | SGOMP | Shifted Gompertz | PDF-S | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

100 | SIL | Siler | Siler Competing Hazards Model | PDF-S | 5 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

101 | SLLOG | Shifted Log-Logistic | Generalized Log-Logistic | PDF-S | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

102 | STACY | Stacy | Distribution Subfamilies: Generalized Gamma; Generalized Inverse
Gamma^{4} |
PDF-S | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

103 | SUNI.D | Step Uniform | Discrete | Discrete | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

104 | T | Student t | Standard | Gosset; Pearson Type IV | PDF-T | 1 | T | t | ● | ● | ● | ● | ● | ● | ● | ● |

105 | T3 | Student t | PDF-T | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

106 | TB | Transformed Beta | PDF-T | 4 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

107 | TRI | Triangular | PDF-T | 3 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

108 | UNI | Uniform | PDF-U | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

109 | UNI.D | Uniform | Discrete | Discrete | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | |

110 | UQ | U-Quadratic | PDF-U | 2 | T | t | ● | ● | ● | ● | ● | ● | ● | ● | ||

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 | ● | ● | ● | ● | ● | ● | ● | ● | |

Notes | ||||||||||||||||

^{1} Probability
Density Function (PDF). |
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^{2} Cumulative
Distribution Function (CDF) (Upper/Lower). Reliability=Upper CDF;
Unreliability=Lower CDF. |
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^{4} PDF Banks:
Continuous PDFs, organized alphabetically in workbooks PDF-A through PDF-Z;
Discrete PDFs in workbook PDF-Discrete. |
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^{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. |
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^{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. |
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^{6} Generalized Extreme Value Distribution (GEV). Wikipedia, 21-Apr-2010,
http://en.wikipedia.org/wiki/Generalized_extreme_value_distribution. |
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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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