# Survival Function / Reliability function

< Probability and statistics definitions < Survival function

What is a survival function?

A survival function (also called a survivor function or reliability function) describes the probability of surviving to a specific time, given that an event of interest has not already occurred. In a more general sense, it’s a function that tells us if an object of interest will survive beyond time t. In medical research, it can be interpreted as the chance an individual will still be alive after a certain age [1].

Survival functions are important in survival analysis, which is used in many fields such as in medicine, where it can model survival times of patients in drug trials. However, these functions aren’t limited to people in ‘survival’ situations. They can be used to describe events for items, objects, or things that have survival-related events. For example, they can be used to model the time between job changes, or the intervals between breakdown events on machines.

A survival distribution is the cumulative distribution function (CDF) of the survival function. In other words, it describes the conditional probability of surviving to a specific time.

## Survival function properties

Given a continuous random variable with CDF F(t) and probability density function (pdf) f(t), the survival function is the integral

S(t) = P(T ≥ t) = ∫[t, ∞] f(x) dx = 1 – F(t)

The survival function of a population of IID random variables T1, · · · , Tn ∼ F can be described as [2]

S(t) = P(T1 > t) = 1 − F(t).

Where F(t) is the CDF. To put this another way, the survival function is the complementary CDF of the lifetime.

Other properties include:

• S(0) = 1and S(∞) = 0.
• S(t) is a non-increasing function (i.e., it can only decrease or stay constant).
• The hazard function is the ratio of the PDF to the survival function, i.e., h(x) = f(x) / S(x) [3].

## References

1. Dwilus, CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0, via Wikimedia Commons
2. Chen, Y. STAT 425: Introduction to Nonparametric Statistics Winter 2018 Lecture 5: Survival Analysis. Retrieved July 22, 2023 from: https://faculty.washington.edu/yenchic/18W_425/Lec5_survival.pdf
3. NIST. Related Distributions.