Survival function

A survival function is a special real function in stochastics that is a supplement to the concept of the distribution function. As with distribution functions, each survival function can be assigned a probability distribution . Conversely, a survival function can be assigned to each probability distribution on the real numbers .

The survival functions take their name from the fact that they occur when modeling lifetimes, for example of individuals or components. If the probability distribution indicates the probability of death of a species, the survival function at that point corresponds to the probability that an individual will be older than . So it “survives” the point in time . A common graph is the survival curve . ${\ displaystyle t}$ ${\ displaystyle t}$ ${\ displaystyle t}$

definition

A probability distribution is given , provided with Borel's σ-algebra , or a real-valued random variable . Then is called ${\ displaystyle P}$ ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle {\ mathcal {B}} (\ mathbb {R})}$ ${\ displaystyle X}$

{\ displaystyle G_ {P} (t): = P ((t, + \ infty))}

respectively

{\ displaystyle G_ {X} (t): = P (X> t)}

the survival function of respectively . ${\ displaystyle P}$ ${\ displaystyle X}$

properties

Similar to the distribution functions, the following applies:

It is and

{\ displaystyle \ lim _ {t \ to - \ infty} G (t) = 1}

{\ displaystyle \ lim _ {t \ to + \ infty} G (t) = 0}

The function is monotonically decreasing ${\ displaystyle G}$
The function is continuous on the right ${\ displaystyle G}$

Relationship to the distribution function

If the distribution function is a probability distribution and the survival function of , then applies ${\ displaystyle F_ {P}}$ ${\ displaystyle P}$ ${\ displaystyle G_ {P}}$ ${\ displaystyle P}$

{\ displaystyle F_ {P} (t) + G_ {P} (t) = 1}

for everyone .

{\ displaystyle t \ in \ mathbb {R}}

The same applies to a random variable ${\ displaystyle X}$

{\ displaystyle F_ {X} (t) + G_ {X} (t) = 1}

for everyone .

{\ displaystyle t \ in \ mathbb {R}}

This follows directly from the definitions of the respective functions and the normalization of the probability distributions. Because the distribution function is precisely the probability of assuming a value less than or equal, the survival function is the probability of assuming a value really greater than . So their sum is the probability of taking on any value and thus one. ${\ displaystyle t}$ ${\ displaystyle t}$

In this way, a distribution function can be obtained from every survival function. A survival function can also be obtained from every distribution function. In particular, analogous to the procedure for distribution functions, each function that fulfills the three points listed under “Properties” can be explained as the survival function of a clearly determined probability distribution (see also correspondence theorem ).

Conditional survival probability and remaining life

If one views a probability distribution as the probability that an individual dies or a component fails, one is often interested in a reassessment of the survival time. If, for example, a quality control has shown that a component is still working at the time , the assessment of the probability will change on the basis of this information. Using the conditional probability , one then obtains the conditional survival probability ${\ displaystyle t_ {0}}$

{\ displaystyle P \ left (X> t_ {0} + t \ mid X> t_ {0} \ right) = {\ frac {G (t + t_ {0})} {G (t_ {0})} } = {\ frac {1-F \ left (t_ {0} + t \ right)} {1-F \ left (t_ {0} \ right)}}}

and for the remaining service life

{\ displaystyle F \ left (t + t_ {0} \ mid t_ {0} \ right) = P \ left (X \ leq t + t_ {0} \ mid X> t_ {0} \ right) = {\ frac {F \ left (t + t_ {0} \ right) -F \ left (t_ {0} \ right)} {1-F \ left (t_ {0} \ right)}}}

Web links

Eric W. Weisstein : Survival Function . In: MathWorld (English).

literature

Klaus D. Schmidt: Measure and Probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , doi : 10.1007 / 978-3-642-21026-6 .