Notation for Distributions

This defines notation for subsequent sections.

Notation

First, let's affix notation. The cumulative distribution function of every regular distribution can be written as an integral over its hazard rate, $\lambda$

\[F(t)=1-e^{-\int_{0}^t \lambda(s)ds}.\]

All algorithms for stochastic simulation treat distributions as being defined in absolute time, specified as an enabling time, $t_e$,

\[F(t, t_e)=1-e^{-\int_{0}^{t-t_e} \lambda(s)ds}.\]

Working with distributions in absolute time is a simple shift of the time scale and will be ignored in further discussions, although the enabling time, $t_e$, will certainly appear in code.

The density function is the derivative of the cumulative distribution function,

\[f(t)=\frac{dF(t)}{dt}=\lambda(t)e^{-\int_{0}^t \lambda(s)ds}.\]

The survival is

\[G(t)=1-F(t)=e^{-\int_{0}^t \lambda(s)ds}.\]

Because survival is multiplicative, we further label the survival from time $t_0$ to $t_1$ as

\[G(t_0, t_1)=\frac{G(t_1)}{G(t_0)}=e^{-\int_{t_0}^{t_1} \lambda(s)ds}\]

Using Julia's Distributions

Julia's continuous univariate distributions support a common interface.

In this table, d is the distribution, t is the time, and q is a quantile.