Cumulative Distribution Function

A cumulative distribution function describes the Probability of a random variable having a value less than or equal to $x$.

A cumulative distribution function is commonly referred to as “at least $x$“.

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Notation

A cumulative distribution function is typically denoted as:

• $P(X\le x)$
• $P(atleastXsuccesses)$

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Formula

The Probability of at least $X$ successes in a given number $n$ trials is: $P(atleastXsuccesses)=1-P(zerosuccesses)-P(onesuccess)-P(...)$

Where

• $P(ysuccesses)$ is given with the Binomial Distribution

The cumulative distribution function for a given continuous random variable is: $P(X\le x)={\int }_{-\infty }^{x}f(t)dt$