# Binomial Distribution

This type of distribution occurs in games of chance, opinion polls, etc. If our aim is to know the times of the occurrence of an event *A* in *n* trials, then *P(A*)=*p* is the success (probability of occurrence of an event A). The probability of not occurring an event *A* is *q* = 1 - p.

Here *X=x* means that *A* occurs in x trials and in *n - x* trials it does not occur. As such, the probability function will be

( *n Л _{n}*!

where 0 < *p* < 1 and I I = which is called the binomial coefficient.

^ x 0* x* !(n - x)!

The distribution function F(x) is defined as
The mean and variance of the binomial distribution are p_{X} = *np* and ct| = *npq.*

# Poisson Distribution

The Poisson distribution is a discrete distribution with the probability function The corresponding distribution function is

This distribution is a special case of binomial distribution, where *p ^* 0 and *n ^* с». The mean and variance of the binomial distribution are p_{X}=*X* and *sX* = l.

# Normal or Gaussian Distribution

This is a continuous distribution and its probability function is defined as

(3.27)

*f ( _{x} ) =*

^{1}

*-r)*

_{e}^{-x}^{2/2s2}

^{f (X)} s/2P ^{e} •

The distribution function of the normal distribution is

* F (x*)

**1**

**af2n**

-(^{x-}m)^{2}/^{2s2}

*dX.*

(3.28)

The integral (3.28) may be written in the closed form as

1 2/2

where Ф *(z*) = 1 *edX* and the mean and variance of the normal distribution is g_{X}=g

2p

and *s ^{2}X =* ct

^{2}, respectively.