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.
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
f (x ) = 1 e-x-r)2/2s2
f (X) s/2P e •
The distribution function of the normal distribution is
The integral (3.28) may be written in the closed form as
where Ф (z) = 1 edX and the mean and variance of the normal distribution is gX=g
and s2X = ct2, respectively.