Page 227 - DMTH404_STATISTICS

Unit 15: Exponential Distribution and Normal Distribution

15.4 Summary Notes

 The random variable in case of Poisson distribution is of the type ; the number of arrivals
of customers per unit of time or the number of defects per unit length of cloth, etc.
Alternatively, it is possible to define a random variable, in the context of Poisson Process,
as the length of time between the arrivals of two consecutive customers or the length of
cloth between two consecutive defects, etc. The probability distribution of such a random
variable is termed as Exponential Distribution.

Let t be a random variable which denotes the length of time or distance between the
occurrence of two consecutive events or the occurrence of the first event and m be the
average number of times the event occurs per unit of time or length. Further, let A be the
event that the time of occurrence between two consecutive events or the occurrence of the
first event is less than or equal to t and f(t) and F(t) denote the probability density function
and the distribution (or cumulative density) function of t respectively.

1
P A
F
( ) .
F
( ) =
 We can write ( ) P A+ ( ) = or ( ) t + P A 1. Note that, by definition, ( ) t = P A
P A
Further, ( ) is the probability that the length of time between the occurrence of two
consecutive events or the occurrence of first event is greater than t. This is also equal to the
probability that no event occurs in the time interval t. Since the mean number of occurrence
of events in time t is mt, we have , by Poisson distribution.
 A large number of chance factors: The factors, affecting the observations of a random
variable, should be numerous and equally probable so that the occurrence or non-
occurrence of any one of them is not predictable.
 Condition of homogeneity: The factors must be similar over the relevant population
although, their incidence may vary from observation to observation.
 Condition of independence: The factors, affecting observations, must act independently of
each other.

 Condition of symmetry: Various factors operate in such a way that the deviations of
observations above and below mean are balanced with regard to their magnitude as well
as their number.
 Normal distribution can also be used to approximate a Poisson distribution when its
parameter m  10. If X is a Poisson variate with mean m, then, for m ³ 10, the distribution
of X can be taken as approximately normal with mean m and standard deviation m so
X m
-
that z = is a standard normal variate.
m
15.5 Keywords

Continuous random variable: A continuous random variable X is said to be uniformly distributed
in a close interval (a, b) with probability density function p(X) if

1
( ) =
p X for a £ X £ b and
-
b a
= 0 otherwise.

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