Use these when life is a count, not a duration: cycles to failure, shocks
survived, demands before a failure. They take the same x/c/n mapping as
the continuous distributions, but there's no probability paper for them — you
get parameters, reliability functions and goodness-of-fit, without the plot.
discrete_weibullq0 – 1β> 0The whole-count analogue of the Weibull, with the same shape-driven behaviour over a discrete support.
R(k) = q^(k^β)
Use it for cycles-to-failure or shocks-to-failure where you'd reach for a Weibull if time were continuous. Watch out: if counts are large (thousands of cycles), the continuous Weibull is usually easier to interpret and loses nothing.
geometricp0 – 1The discrete memoryless model — the counting analogue of the exponential. Each trial fails independently with the same probability.
R(k) = (1 − p)^k
Use it for per-demand failures where history genuinely doesn't matter. Watch out: as with the exponential, the memoryless assumption is often assumed rather than checked — if failures cluster with age, this will hide it.
beta_geometrica> 0b> 0Geometric with unit-to-unit variation. The per-trial failure probability isn't one number: it varies across the population, drawn from a Beta.
p ~ Beta(α, β), then k | p ~ Geometric(p)
Use it when a plain geometric under-predicts survivors — heterogeneity means weak units fail early and the remainder are tougher than the average implies. Watch out: you need a decent sample to separate real heterogeneity from noise.
negative_binomialr> 0p0 – 1Trials until several failures, rather than the first — more dispersed than the geometric.
Number of trials until the r-th failure, each with probability p
Use it for counts that are over-dispersed relative to a simple model, or when a unit tolerates several events before it's considered failed. Watch out: its extra parameter can absorb over-dispersion that actually comes from mixing distinct populations.
poissonμ> 0Counts of events in a fixed exposure — the classic count model.
P(k) = e^(−λ)·λ^k / k!
λ is both the mean and the variance.
Use it for number of events per unit of time or usage. Watch out: the mean-equals-variance property is a strong claim. If your data's variance clearly exceeds its mean, prefer the negative binomial. Note too that this models a count, not a time to failure — for repairable-system event histories, the recurrent-event models are the right tool.