Kertymäfunktions

Kertymäfunktions is a term associated with the cumulative distribution function (CDF) of a random variable, often known in Finnish as kertymäfunktio. It maps each real number x to the probability that the variable takes a value less than or equal to x: F(x) = P(X ≤ x). As a fundamental object in probability and statistics, it characterizes the distribution of a random variable completely.

Key mathematical properties include monotonic non-decrease (if a ≤ b then F(a) ≤ F(b)), right-continuity, and boundary limits lim_{x→−∞} F(x) = 0 and lim_{x→∞} F(x) = 1. For absolutely continuous distributions, the derivative F′(x) exists almost everywhere and equals the probability density function (pdf). For discrete distributions, F has jump discontinuities, with jump sizes equal to point probabilities.

Kertymäfunktions are used to compute probabilities for intervals, derive moments, and define related functions such as the survival function S(x) = 1 − F(x) and the quantile (inverse CDF) function used in sampling methods like the inverse transform. The empirical distribution function is the sample-based estimator of the true CDF and underlies consistency results such as the Glivenko–Cantelli theorem.

Generalizations include joint cumulative distribution functions for multivariate variables and conditional distribution functions. Applications span statistical inference, reliability and survival analysis, risk assessment, and stochastic modeling.