## Binomial distribution

The probability of binomial distribution, in which the probability of a single trial of event is the $p$ and events occurs $x$ times in n trials, is given by the following equation.
$p(x)=\left(

\begin{array}{c}

n \\

x \\

\end{array}

\right)p^x (1-p)^{n-x} , 0\leq x\leq n$

\begin{array}{c}

n \\

x \\

\end{array}

\right)p^x (1-p)^{n-x} , 0\leq x\leq n$

(1)

where,

$\left(

\begin{array}{c}

n \\

x \\

\end{array}

\right)=\frac{n!}{x! (n-k)!}$

\begin{array}{c}

n \\

x \\

\end{array}

\right)=\frac{n!}{x! (n-k)!}$

(2)

Binomial distribution random numbers can be obtained by standard exponential distribution random number sequence $E_1, E_2,\cdots$. Maximum of $N$ satisfying the following formula is binomially distributed.

$\sum _{i=1}^N \frac{E_i}{m-i+1}\leq -\log (1-p)$

(3)

Reference:

- M.Fushimi, Random number, UP Sensho Applied Mathematics, University of Tokyo Press, 1989

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