Cumulative distribution function

In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable , or just distribution function of , evaluated at , is the probability that will take a value less than or equal to . Every probability distribution supported on the real numbers, discrete or "mixed" as well as continuous, is uniquely identified by a right-continuous monotone increasing function (a càdlàg function) satisfying and . In the case of a scalar continuous distribution, it gives the area under the probability density function from minus infinity to . Cumulative distribution functions are also used to specify the distribution of multivariate random variables. The cumulative distribution function of a real-valued random variable is the function given by where the right-hand side represents the probability that the random variable takes on a value less than or equal to . The probability that lies in the semi-closed interval , where , is therefore In the definition above, the "less than or equal to" sign, "≤", is a convention, not a universally used one (e.g. Hungarian literature uses "