Binomial Distribution

A binomial random variable with parameters \(\left(n,p\right)\) can be described as the sum of \(n\) independent Bernoulli random variables of parameter \(p;\)

\[ Y=\sum_{i=1}^{n}X_{i}.\]

Therefore, this random variable counts the number of successes in \(n\) independent trials of a random experiment where the probability of success is \(p.\)

\[ \begin{eqnarray*} p\left(k;n,p\right) & = & \left(\begin{array}{c} n\\ k\end{array}\right)p^{k}\left(1-p\right)^{n-k}\,\, k\in\left\{ 0,1,\ldots n\right\} ,\\ F\left(x;n,p\right) & = & \sum_{k\leq x}\left(\begin{array}{c} n\\ k\end{array}\right)p^{k}\left(1-p\right)^{n-k}=I_{1-p}\left(n-\left\lfloor x\right\rfloor ,\left\lfloor x\right\rfloor +1\right)\quad x\geq0\end{eqnarray*}\]

where the incomplete beta integral is

\[ I_{x}\left(a,b\right)=\frac{\Gamma\left(a+b\right)}{\Gamma\left(a\right)\Gamma\left(b\right)}\int_{0}^{x}t^{a-1}\left(1-t\right)^{b-1}dt.\]

Now

\[ \begin{eqnarray*} \mu & = & np\\ \mu_{2} & = & np\left(1-p\right)\\ \gamma_{1} & = & \frac{1-2p}{\sqrt{np\left(1-p\right)}}\\ \gamma_{2} & = & \frac{1-6p\left(1-p\right)}{np\left(1-p\right)}.\end{eqnarray*}\]

\[ M\left(t\right)=\left[1-p\left(1-e^{t}\right)\right]^{n}\]

Implementation: scipy.stats.binom