Methods and formulas for distribution functions in Parametric Distribution Analysis (Right Censoring)

Pdf and cdf

The probability density function (pdf) and cumulative distribution function (cdf) for each reliability distribution are as follows:

Smallest extreme value

pdf
cdf

Weibull

pdf
cdf

3-parameter Weibull

pdf
cdf

Exponential

pdf
cdf

2-parameter exponential

pdf
cdf

Normal

pdf
cdf

Lognormal

pdf
cdf

3-parameter lognormal

pdf
cdf

Logistic

pdf
cdf

Loglogistic

pdf
cdf

3-parameter loglogistic

pdf
cdf

For more information on distribution parameters, see the section "Parameter estimates".

Notation

TermDescription
μ location parameter
σ scale parameter (Weibull and 3-parameter Weibull)
θ scale parameter (smallest extreme value, exponential, and 2-parameter exponential)
αscale parameter (other distributions)
β shape parameter
λ threshold parameter

Hazard function

The hazard function provides a measure of the likelihood of failure as a function of how long a unit has survived (the instantaneous failure rate at a particular time, t).

Formula

Notation

TermDescription
f(t) pdf of the chosen distribution
F(t) cdf of the chosen distribution

Relationships between distributions

When data follow a log-based distributions (Weibull, exponential, lognormal, loglogistic), sometimes calculations are simpler by first taking the log of your data and fitting the corresponding related distribution.

If your data follow this distribution... The log of your data will follow this distribution...
Weibull, exponential Smallest extreme value
Lognormal Normal
Loglogistic Logistic

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