Binomial distribution
- $N \in \mathbb{N}$,
- given
- $p \in [0, 1]$,
- given
- probability
- \(X_{i} \in \{0, 1\} \ (i = 1, \ldots, N)\),
- i.i.d sequence of Bernoulli r.v.
- $P(X_{i} = 0) = p \ (1, \ldots, N)$,
$\mathrm{Bi}(x_{1}, \ldots, x_{N}; p)$ is a p.d.f. of binomial distribution given $p$ and $N$ defined by
\[\mathrm{Bi}(x_{1}, \ldots, x_{N}; p) := \left( \begin{array}{c} N \\ \sum_{i=1}^{N} x_{i} \end{array} \right) p^{ \sum_{i=1}^{N} x_{i} } (1 - p)^{ N - \sum_{i=1}^{N} x_{i} } .\] \[\begin{eqnarray} F_{\mathrm{Bi}}(n; p) & := & \sum_{x \in } \mathrm{Bi}(x_{1}, \ldots, x_{n}; p) \nonumber \\ n \in \{0, 1, \ldots, N\}, \ F_{\mathrm{Bi}}(n; p) & := & \sum_{x \in } \mathrm{Bi}(x_{1}, \ldots, x_{n}; p) \end{eqnarray}\]