Table 2.

Summary of Continuous-Time Probability Models for the Transition Probability Density Function p(τ) (Where τ Is a Nonnegative or Positive Random Variable That Denotes the Holding Time), Cumulative Distribution Function F(τ), and Survival Function 1 − F(τ).

	pdf p(τ)	cdf F(τ)	l– F(τ)	Domain
Exponential	r exp(−rτ)	1 − exp(−rτ)	exp(−rτ)	τ ∈ [0, ∞), r > 0
Weibull	rα(rτ)α−1 exp[−(rτ)α]	1 − exp[−(rτ)α]	exp[−(rτ)α]	τ ∈ [0, ∞), r > 0, α > 0
Gamma	${\tau }^{s-1}\frac{exp(-\tau /\theta )}{\mathrm{\Gamma }(s){\theta }^s}$	$\frac{\gamma (s,\tau /\theta )}{\mathrm{\Gamma }(s)}$	$1-\frac{\gamma (s,\tau /\theta )}{\mathrm{\Gamma }(s)}$	τ ∈ [0, ∞), s > 0, θ > 0
Log normal	$\frac{1}{\tau \sqrt{2\pi {\sigma }^2}}exp(-\frac{{(ln\tau -\mu )}^2}{2{\sigma }^2})$	$\frac{1}{2}+\frac{1}{2}\text{erf}[\frac{ln\tau -\mu }{\sqrt{2}\sigma }]$	$\frac{1}{2}-\frac{1}{2}\text{erf}[\frac{ln\tau -\mu }{\sqrt{2}\sigma }]$	τ ∈ (0, ∞), μ > 0, σ > 0
Inverse gaussian	$\sqrt{\frac{s}{2\pi {\tau }^3}}exp(-\frac{s{(\tau -\mu )}^2}{2{\mu }^2\tau })$	$\mathrm{\Phi }(\sqrt{\frac{s}{\tau }}(\frac{t}{\mu }-1))+exp(\frac{2s}{\tau })\mathrm{\Phi }(-\sqrt{\frac{s}{\tau }}(\frac{t}{\mu }+1))$	–	τ ∈ (0, ∞), μ > 0, s > 0

Note: $\text{erf}(z)=\frac{2}{\sqrt{\pi }}{\int }_0^{z}exp(-t^2)dt$ denotes the error function, $\mathrm{\Phi }(z;\mu ,\sigma )={\int }_{-\infty }^{z}exp(-\frac{{(t-\mu )}^2}{2{\sigma }^2})dt$ denotes the gaussian cumulative distribution function, and these two functions relate to each other by $\mathrm{\Phi }(z)=\frac{1}{2}[1+\text{erf}(\frac{z}{\sqrt{2}})]$.