Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Univariate discrete distributions ▷ Binomial distribution ▷ Cumulative distribution function

Theorem: Let $X$ be a random variable following a binomial distribution:

\[\label{eq:bin} X \sim \mathrm{Bin}(n,p) \; .\]

Then, the cumulative distribution function of $X$ is

\[\label{eq:bin-cdf} F_X(x) = \sum_{k=0}^{x} {n \choose k} \, p^k \, (1-p)^{n-k} \; .\]

Proof: The cumulative distribution function is defined as

\[\label{eq:cdf} F_X(x) = \mathrm{Pr}(X \leq x)\]

which, in terms of the probability mass function, is given by

\[\label{eq:cdf-pmf} F_X(x) = \sum_{\substack{t \in \mathcal{X} \\ t \leq x}} f_X(t) \; .\]

The probability mass function of the binomial distribution is

\[\label{eq:bin-pmf} f_X(x) = {n \choose x} \, p^x \, (1-p)^{n-x} \; ,\]

so that the cumulative distribution function of the binomial distribution becomes

\[\label{eq:bin-cdf-qed} \begin{split} F_X(x) &\overset{\eqref{eq:cdf-pmf}}{=} \sum_{k=0}^{x} f_X(k) \\ &\overset{\eqref{eq:bin-pmf}}{=} \sum_{k=0}^{x} {n \choose k} \, p^k \, (1-p)^{n-k} \; . \end{split}\]
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Metadata: ID: P488 | shortcut: bin-cdf | author: JoramSoch | date: 2025-02-06, 14:37.