In probability theory and statistics, the discrete Weibull distribution is the discrete variant of the Weibull distribution. The Discrete Weibull Distribution, first introduced by Toshio Nakagawa and Shunji Osaki, is a discrete analog of the continuous Weibull distribution, predominantly used in reliability engineering. It is particularly applicable for modeling failure data measured in discrete units like cycles or shocks. This distribution provides a versatile tool for analyzing scenarios where the timing of events is counted in distinct intervals, making it distinctively useful in fields that deal with discrete data patterns and reliability analysis. The discrete Weibull distribution is infinitely divisible only for ${\displaystyle 0<\beta \leq 1}$.

In the original paper by Nakagawa and Osaki they used the parametrization ${\displaystyle p=q^{k^{-\beta }}}$ making the cumulative distribution function ${\displaystyle 1-q^{(x+1)^{\beta }}}$

with ${\displaystyle q\in (0,1)}$ and the probability mass function ${\displaystyle q^{k^{\beta }}-q^{(k+1)^{\beta }}}$

. Setting ${\displaystyle \beta =1}$ makes the relationship with the geometric distribution apparent.

An alternative parametrization — related to the Pareto distribution — has been used to estimate parameters in infectious disease modelling. This parametrization introduces a parameter ${\displaystyle \kappa ={\frac {\beta }{\alpha ^{\beta }}}}$, meaning that the term ${\displaystyle \left({\frac {1}{\alpha }}\right)^{\beta }}$ can be replaced with ${\displaystyle {\frac {\kappa }{\beta }}}$. Therefore, the probability mass function can be expressed as

and the cumulative mass function can be expressed as

The continuous Weibull distribution has a close relationship with the Gumbel distribution which is easy to see when log-transforming the variable. A similar transformation can be made on the discrete Weibull.

Define ${\displaystyle e^{Y}-1=X}$ where (unconventionally) ${\displaystyle Y=\log(X+1)\in \{\log(1),\log(2),\ldots \}}$ and define parameters ${\displaystyle \mu =\log(\alpha )}$ and ${\displaystyle \sigma ={\frac {1}{\beta }}}$. By replacing ${\displaystyle x}$ in the cumulative mass function: