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Survival function

The survival function, also known as a survivor function or reliability function, is a property of any random variable that maps a set of events, usually associated with mortality or failure of some system, onto time. It captures the probability that the system will survive beyond a specified time. The term reliability function is common in engineering while the term survival function is used in a broader range of applications, including human mortality.

Definition

Let X be a continuous random variable with cumulative distribution function F(t) on the interval [0,∞). Its survival-, or reliability-function is:

$R(t) = P(\{T > t\}) = \int_t^{\infty} f(u)\,du = 1-F(t).$

Properties

Every survival function R(t) is monotone decreasing, i.e. R(u) < R(t) for u > t

The time, t = 0, represents some origin, typically the beginning of a study or the start of operation of some system. R(0) is commonly unity but can be less to represent the probability that the system fails immediately upon operation.