📂Functions

Survival Function

Definition ¹

If a function $S(0)=1$ is non-increasing, it is defined as a Survival Function.

Description

The survival function, in simple terms, is a function that maps the probability $S(t) \in [0,1]$ of being alive at time $t$. In mathematics, survival doesn’t necessarily stick to the meaning of ‘being alive’ but is abstracted to the period until a certain event occurs, and since it’s about mapping probabilities, it naturally has a relationship with the probability density function. If the point at which survival ends is denoted by the random variable $T$, then, with respect to the cumulative distribution function $F(t)$, $$ F(t) = P (T \le t) = 1 - S(t) $$ and with respect to the probability density function $f$, $$ f(t) = {{ d } \over { dt }} F(t) $$ it can be expressed. Hence, the survival function is sometimes derived inversely from $T$’s probability density function. This is mostly how it is used, and the definition introduced above is written as generally as possible without even depending on probability.

The distributions mainly used in survival analysis include the exponential distribution and gamma distribution, Weibull distribution, etc.