Cumulative Distribution Function of Binomial Distribution

Theorem

The cumulative distribution function of a binomially distributed random variable $X$ is equal to:

$\map \Phi x = \ds \sum_{t \mathop \le x} \dbinom n t p^t \paren {1 - p}^{n - t}$

where:

$n$ is the number of trials
$p$ is the probability of success such that $0 \le p \le 1$.


Proof


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Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 39$: Probability Distributions: Binomial Distribution: $39.1$
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