Cumulative distribution function
One class of binomial probabilities is particularly important. The probability that the random variable \(X\) is less than or equal to a constant \(x\) is its cumulative probability,
\[ F(x) = P(X \le x) = \sum_{u \le x} {p(u)} = \sum_{u=0}^{\lfloor x \rfloor} {{n \choose u}\pi^x (1-\pi)^{n-x} }\]where \(\lfloor x \rfloor\) denotes the largest integer less than or equal to \(x\). Note that, unlike the probability function \(p(x)\), this is defined for all \(x\), not just for integer values.
Unfortunately there is no simple formula for the cumulative probabilities of the binomial distribution.
When treated as a function of \(x\), this is the distribution's cumulative distribution function (CDF) — a step function that increases by \(p(x)\) at each integer value from 0 to \(n\).
Illustration
The bottom half of the following diagram shows the bar chart for a binomial distribution. Above it is the distribution's CDF.
Notice that the larger probabilities correspond to higher steps in the CDF.
Adjust the probability of success, \(\pi\), and observe how the shape of the CDF changes to reflect any skewness in the distribution.
Adjust the sample size, \(n\), and observe that the shape of the CDF becomes smoother when \(n\) becomes larger, approaching a smooth curve.