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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 pX = 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 pX=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 gX=g

2p

and s2X = ct2, respectively.

 
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