The R Book

The Exponential Distribution

This is a one-parameter distribution in which the hazard function is independent of age (i.e. it describes a Type II survivorship curve). The exponential is a special case of the gamma distribution in which the shape parameter α is equal to 1.

Density function

The density function is the probability of dying in the small interval of time between t and t + dt, and a plot of the number dying in the interval around time t as a function of t (i.e. the proportion of the original cohort dying at a given age) declines exponentially:

where both μ and t > 0. Note that the density function has an intercept of 1/μ (remember that e⁰ is 1). The number from the initial cohort dying per unit time declines exponentially with time, and a fraction 1/μ dies during the first time interval (and, indeed, during every subsequent time interval).

Survivor function

This shows the proportion of individuals from the initial cohort that are still alive at time t:

The survivor function has an intercept of 1 (i.e. all the cohort is alive at time 0), and shows the probability of surviving at least as long as t.

Hazard function

This is the statistician's equivalent of the ecologist's instantaneous death rate. It is defined as the ratio between the density function and the survivor function, ...