Definition#

The cumulative distribution function (CDF) of a real-valued random variable $X$ is the function given by:[1]

where the right-hand side represents the probability that the random variable $X$ takes on a value less than or equal to $x$.

The probability that $X$ lies in the semi-closed interval $(a, b]$, where $a < b$, is therefore:[1]

The CDF of a continuous random variable $X$ can be expressed as the integral of its probability density function (PDF) $f_X$ as follows:[1]

CDF for the normal distribution ${\mathcal{N}}(\mu ,\sigma ^{2})$#