Page 291 - DCOM203_DMGT204_QUANTITATIVE_TECHNIQUES_I

Quantitative Techniques – I

Notes n r
m r lim 1 m
. n n 1 n 2 .... n r 1 . r . 1
! r n n n

n 1 2 r 1 m n
1 1 .... 1 1
r n n n n n
m lim
= . r
r! n m
1
n
r n
m m
lim 1 since each of the remaining terms will tend to unity as n
! r n
n

m
n
n
r
m .e -m lim m lim m m
since 1 1 e m
! r n n n n
Thus, the probability mass function of Poisson distribution is
m r
e .m
P r , where r 0,1,2, ......
! r
Here e is a constant with value = 2.71828... . Note that Poisson distribution is a discrete probability
distribution with single parameter m.

m r 2 3
e .m m m m m
Total probability e 1 ....
! r 1! 2! 3!
r 0
m m
e .e 1
14.1.2 Summary Measures of Poisson Distribution

1. Mean: The mean of a Poisson variate r is defined as

m r r 3 4
e .m m m m 2 m m
E r . r e e m m ....
! r r 1 ! 2! 3!
r 0 r 1
2 3
m m m m m
me 1 m .... me e m
2! 3!
2. Variance: The variance of a Poisson variate is defined as
2
2
Var(r) = E(r - m) = E(r ) - m 2
2
Now E r 2 r P r r r 1 r P r
r 0 r 0

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