Survival function
Encyclopedia
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.
F(t) on the interval[0,∞) . Its survival-, or reliability-function is:
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.
The survivor function is right continuous.
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
that maps a set of events, usually associated with mortality or failure of some system, onto time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
. It captures the probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
that the system will survive
Survival analysis
Survival analysis is a branch of statistics which deals with death in biological organisms and failure in mechanical systems. This topic is called reliability theory or reliability analysis in engineering, and duration analysis or duration modeling in economics or sociology...
beyond a specified time. The term reliability function is common in engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...
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 functionCumulative distribution function
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...
F(t) on the interval
Properties
Every survival function R(t) is monotone decreasing, i.e. for all .The time
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
, 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
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
that the system fails immediately upon operation.
The survivor function is right continuous.
See also
- Survival analysisSurvival analysisSurvival analysis is a branch of statistics which deals with death in biological organisms and failure in mechanical systems. This topic is called reliability theory or reliability analysis in engineering, and duration analysis or duration modeling in economics or sociology...
- Failure rateFailure rateFailure rate is the frequency with which an engineered system or component fails, expressed for example in failures per hour. It is often denoted by the Greek letter λ and is important in reliability engineering....
- Mean time to failureMean time to failureNo artical exists on Wiki, please create one.In short Mean Time to Failure is the time taken for a part or system to fail for the first time.A very brief formula for the Mean Time To Failure of an event which occurs with probability P is: 1 / P....
- Kaplan–Meier estimator