Survivor Function¶

Definition¶

For a random variable $X$ with CDF $F$ , the survivor function is

\begin{aligned} \overline{F}(x) = 1 - F(x) = \mathbb{P}(X > x). \end{aligned}

Interpretation¶

The survivor function gives the probability of exceeding a threshold. For a lifetime or a waiting time $X$ , $\overline F(x)$ is the probability of still waiting, or still working, at time $x$ ; it is useful for reliability and tail probabilities.

For the Exponential(1) distribution, the increasing CDF F(x) and the decreasing survivor function 1 - F(x), with their values at x = 1, about 0.63 and 0.37, marked.

The illustrated $\operatorname{Exp}(1)$ distribution has survivor function $\overline F(x)=\mathrm{e}^{-x}$ for $x\ge0$ $0$ . At $x=1$ the two curves take the values $F(1)\approx0.63$ and $\overline F(1)\approx0.37$ $0.37$ , which add up to $1$ , as they do at every $x$ . For rate $\lambda>0$ $0$ , the survivor function becomes $\mathrm{e}^{-\lambda x}$ .

Many waiting-time facts are simplest in terms of the tail. The memoryless property of the exponential distribution reads $\overline F(s+t)=\overline F(s)\overline F(t)$ $\overline{F} (s) \overline{F} (t)$ , and for independent variables the minimum exceeds $x$ exactly when every variable does, so its survivor function is the product of theirs, as for the minimum of independent exponentials.

The inequality is strict: $\overline F(x)=\mathbb{P}(X>x)$ $x)$ . For continuous variables this equals $\mathbb{P}(X\ge x)$ $x)$ , but for discrete variables the two differ by $\mathbb{P}(X=x)$ $x)$ .

Key Results¶

$\overline{F}(x)=1-F(x)$ is nonincreasing and right-continuous.
For a discrete variable, $\mathbb{P}(X\ge x)=1-F(x-)$ $F (x -)$ , which can differ from $\overline{F}(x)$ .

Examples¶

For $\operatorname{Exp}(3)$ , $\overline{F}(2)=\mathrm{e}^{-6}$ $e^{- 6}$ .
For $\operatorname{Bern}(0.4)$ , $\mathbb{P}(X>0)=0.4$ but $\mathbb{P}(X\ge0)=1$ $1$ .

Prerequisites¶

Cumulative Distribution Function

Sources¶

Nicky van Foreest · Probability Distributions · Week 2

Aliases: survival function, reliability function, complementary cumulative distribution function, complementary CDF, CCDF, tail distribution function Symbols: lambda

Last updated: 20 October 2026 03:02