Page 196 - DMTH404_STATISTICS

Statistics

Notes 14.3 Pascal Distribution

In binomial distribution, we derived the probability mass function of the number of successes in
n (fixed) Bernoulli trials. We can also derive the probability mass function of the number of
Bernoulli trials needed to get r (fixed) successes. This distribution is known as Pascal distribution.
Here r and p become parameters while n becomes a random variable.

We may note that r successes can be obtained in r or more trials i.e. possible values of the
random variable are r, (r + 1), (r + 2), ...... etc. Further, if n trials are required to get r successes, the
nth trial must be a success. Thus, we can write the probability mass function of Pascal distribution
as follows:

æ Probability of ( r  ) 1 successesö æ Probability of a successö
P ( ) n = ç ÷ ´ ç ÷
è out of ( n  ) 1 trials ø è in nth trial ø

r n r

= n 1 C p r 1 n r ´ p = n 1 C p q  , where n = r, (r + 1), (r + 2), ... etc.
q
r 1 r 1
r rq
It can be shown that the mean and variance of Pascal distribution are and respectively.
p p 2
This distribution is also known as Negative Binomial Distribution because various values of
P(n) are given by the terms of the binomial expansion of p (1 - q) .
- r
r
14.4 Geometrical Distribution

When r = 1, the Pascal distribution can be written as
P ( ) n = n 1 C pq n 1 = pq n 1 , where n = 1,2,3,.....
0
Here n is a random variable which denotes the number of trials required to get a success. This
distribution is known as geometrical distribution. The mean and variance of the distribution are
1 q
and respectively.
p p 2

14.5 Uniform Distribution (Discrete Random Variable)

A discrete random variable is said to follow a uniform distribution if it takes various discrete
values with equal probabilities.
1
If a random variable X takes values X , X , ...... X each with probability , the distribution of X
1 2 n n
is said to be uniform.
14.6 Poisson Distribution

This distribution was derived by a noted mathematician, Simon D. Poisson, in 1837. He derived
this distribution as a limiting case of binomial distribution, when the number of trials n tends to
become very large and the probability of success in a trial p tends to become very small such that
their product np remains a constant. This distribution is used as a model to describe the probability
distribution of a random variable defined over a unit of time, length or space. For example, the
number of telephone calls received per hour at a telephone exchange, the number of accidents in

188 LOVELY PROFESSIONAL UNIVERSITY