Editing Survival analysis (section)

===Lifetime distribution function and event density===

Related quantities are defined in terms of the survival function.

The '''lifetime distribution function''', conventionally denoted ''F'', is defined as the complement of the survival function,

<math display="block">F(t) = \Pr(T \le t) = 1 - S(t).</math>
If ''F'' is [[Differentiable function|differentiable]] then the derivative, which is the density function of the lifetime distribution, is conventionally denoted ''f'',

<math display="block">f(t) = F'(t) = \frac{d}{dt} F(t).</math>
The function ''f'' is sometimes called the '''event density'''; it is the rate of death or failure events per unit time.

The survival function can be expressed in terms of [[probability distribution]] and [[probability density function]]s

<math display="block">S(t) = \Pr(T > t) = \int_t^{\infty} f(u)\,du = 1-F(t).</math>
Similarly, a survival event density function can be defined as

<math display="block">s(t) = S'(t) = \frac{d}{dt} S(t) = \frac{d}{dt} \int_t^{\infty} f(u)\,du = \frac{d}{dt} [1-F(t)] = -f(t).</math>
In other fields, such as statistical physics, the survival event density function is known as the [[first passage time]] density.