Binomial distribution

• $N\in \mathbb{N}$,
  • given
• $p\in [0,1]$,
  • given
  • probability
• $X_i\in \{0,1\}(i=1,\dots ,N)$,
  • i.i.d sequence of Bernoulli r.v.
  • $P(X_i=0)=p(1,\dots ,N)$,

$\mathrm{Bi}(x_1,\dots ,x_N;p)$ is a p.d.f. of binomial distribution given $p$ and $N$ defined by

$$\mathrm{Bi}(x_1,\dots ,x_N;p):=\left(\begin{array}{c}N\\ \sum _{i=1}^N{x}_i\end{array}\right)p^{\sum _{i=1}^N{x}_i}(1-p{)}^{N-\sum _{i=1}^N{x}_i}.$$ $$\begin{array}{rcl}{F}_{\mathrm{Bi}}(n;p)& :=& \sum _{x\in }\mathrm{Bi}(x_1,\dots ,x_n;p)\\ \text{(1)}& n\in \{0,1,\dots ,N\},F_{\mathrm{Bi}}(n;p)& :=& \sum _{x\in }\mathrm{Bi}(x_1,\dots ,x_n;p)\end{array}$$

Reference