# 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.

Julia call | Notation |
---|---|

`cdf(d,t)` | $F(t)$ |

`quantile(d,q)` | $F^{-1}(q)$ |

`logcdf(d,t)` | $\ln(F(t))$ |

`ccdf(d,t)` | $G(t)$ |

`logccdf(d,t)` | $-\int_0^t \lambda(s)ds$ |

`quantile(d,q)` | $F^{-1}(q)$ |

`cquantile(d,q)` | $F^{-1}(1-q)=G^{-1}(q)$ |

`invlogcdf(d,lp)` | $F^{-1}(e^{l_p})$ |

`invlogccdf(d,lp)` | $G^{-1}(e^{l_p})$ or $-\int_0^{t(l_p)}\lambda(s)ds=l_p$ |

`randexp(rng)` | $-\ln(1-U)$ |