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}_{e}^{-x}-r)^{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