For questions related to cumulative distribution functions.
The cumulative distribution function $F_X: \Bbb R \to [0,1]$ of a real-valued random variable $X$ is the probability that random variable $X$ will take a value less than $x$. Given the probability distribution of $X$ given by $f_X:\mathbb R\to[0,1]$, this is formally written as: $$F_X(x) := \Bbb P (X \leq x)=\int_{-\infty}^xf_Xdx$$
Every CDF is non-decreasing, right-continuous, and satisfies $\lim_{x\to-\infty}F_X=0$ and $\lim_{x\to\infty}F_X=1$. These conditions are sufficient to be a CDF, as any function satisfying these $4$ conditions is a CDF of some probability distribution.
The concept of the cumulative distribution function makes an explicit appearance in statistical analysis in two (similar) ways.
- Cumulative frequency analysis is the analysis of the frequency of occurrence of values of a phenomenon less than a reference value.
- The empirical distribution function is a formal direct estimate of the cumulative distribution function for which simple statistical properties can be derived and which can form the basis of various statistical hypothesis tests.
Such tests can assess whether there is evidence against a sample of data having arisen from a given distribution or evidence against two samples of data having arisen from the same (unknown) population distribution.
