In statistics and probability, the **cumulative distribution function**, abbreviated **cdf**, of a real-valued variable *X* is a function that, when evaluated at *X=x,* gives the probability that X has a value less than or equal to *x*. By convention, cumulative distribution functions are denoted by upper case letters, often *F*. The cdf *F* for random variable *X *would be . So, by definition:

For a discrete probability distribution over *X*, the cumulative distribution function at *X=x* is the sum of the probability mass function values of *X* starting with the first *X* having non-zero probability up to and including *x*.

If the random variable *X* is represented by a continuous distribution, the cumulative distribution function at *X=x* gives the area under the probability density function from minus infinity to *x*. This is equivalent to the integral of the probability distribution function from minus infinity to x:

The cumulative distribution function is non-decreasing and right-continuous. Because probabilities must be between 0 and 1, and the totally probability of sample space is 1:

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