Page 209 - DMTH404_STATISTICS

Unit 15: Exponential Distribution and Normal Distribution

Theoretical probability distributions can be divided into two broad categories, viz. discrete and Notes
continuous probability distributions, depending upon whether the random variable is discrete
or continuous. Although, there are a large number of distributions in each category, we shall
discuss only some of them having important business and economic applications.

15.1 Exponential Distribution

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.
Since the length of time or distance is a continuous random variable, therefore exponential
distribution is a continuous probability distribution.

15.1.1 Probability Density Function

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.

We can write P A a f+P A d i =1 or F t a f+P A d i =1. Note that, by definition, F t a f = P A a f .
Further, P A d i 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,

e - mt ( ) 0
mt
( ) P=
P A (r = ) 0 = = e - mt .
0!
-mt
Thus, we get F(t) + e = 1
-mt
or P(0 to t) = F(t) = 1 - e . .... (1)
To get the probability density function, we differentiate equation (1) with respect to t.
Thus, f(t) = F'(t) = me when t > 0
-mt
= 0 otherwise.
It can be verified that the total probability is equal to unity

¥
¥ e - mt ¥
+
=
Total Probability = ò m .e - mt dt = m . = - e - mt = 0 1 1.
0 - m 0
0

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