information:filtering
Complementary cumulative distribution function (tail distribution)

Sometimes, it is useful to study the opposite question and ask how often the random variable is above a particular level. This is called the complementary cumulative distribution function (ccdf) or simply the tail distribution or exceedance, and is defined as F ¯ X (x) = P (X > x) = 1 − F X (x). {\displaystyle {\bar {F}}_{X}(x)=\operatorname {P} (X>x)=1-F_{X}(x).}

This has applications in statistical hypothesis testing, for example, because the one-sided p-value is the probability of observing a test statistic at least as extreme as the one observed. Thus, provided that the test statistic, T, has a continuous distribution, the one-sided p-value is simply given by the ccdf: for an observed value t {\displaystyle t} of the test statistic p = P (T ≥ t) = P (T > t) = 1 − F T (t). {\displaystyle p=\operatorname {P} (T\geq t)=\operatorname {P} (T>t)=1-F_{T}(t).}

In survival analysis, F ¯ X (x) {\displaystyle {\bar {F}}_{X}(x)} is called the survival function and denoted S (x) {\displaystyle S(x)}, while the term reliability function is common in engineering.