## Survival analyses

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. Another name for the survival function is the **complementary cumulative distribution function**.

## Definition

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

## Properties

Every survival function *S*(*t*) is monotonically decreasing, i.e. for all .

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.

Since the CDF is a right-continuous function and the survival function, , it can be said that the survival function is also right-continuous.

## Kaplan-Meier estimates

The **Kaplan–Meier estimator**,^{[1]}^{[2]} also known as the **product limit estimator**, is an estimator for estimating the survival function from lifetime data. In medical research, it is often used to measure the fraction of patients living for a certain amount of time after treatment. In economics, it can be used to measure the length of time people remain unemployed after a job loss. In engineering, it can be used to measure the time until failure of machine parts. In ecology, it can be used to estimate how long fleshy fruits remain on plants before they are removed by frugivores. The estimator is named after Edward L. Kaplan and Paul Meier.

*I realize the source is mostly Wikipedia, but I created the post for my personal needs, hopefully with making a proper post on my blog in the near future*.