Survival function

The term reliability function is common in engineering while the term survival function is used in a broader range of applications, including human mortality. The survival function is the complementary cumulative distribution function of the lifetime. Sometimes complementary cumulative distribution functions are called survival functions in general. Let the lifetime T {\displaystyle T} be a continuous random variable describing the time to failure. If T {\displaystyle T} has cumulative distribution function F ( t ) {\displaystyle F(t)} and probability density function f ( t ) {\displaystyle f(t)} on the interval [ 0 , ∞ ) {\displaystyle [0,\infty )} , then the survival function or reliability function is: S ( t ) = ∫ t ∞ f ( u ) d u = Pr ( T > t ) = 1 − F ( t ) = 1 − ∫ 0 t f ( u ) d u {\displaystyle S(t)=\int _{t}^{\infty }f(u)\,du=\Pr(T>t)=1-F(t)=1-\int _{0}^{t}f(u)\,du} The graphs below show examples of hypothetical survival functions. The x-axis is time. The y-axis is the proportion of subjects surviving. The graphs show the probability that a subject will survive beyond time t.