Page 199 - DMTH404_STATISTICS

Unit 14: Theoretical Probability Distributions

Notes
 e  m .m r  m r
= å é ë ( r r  ù û + m = e  m å + m
) 1
r= 2 ! r r= 2 (r  ) 2 !

æ m 4 m 5 ö
2
3
m e= +  m m + m + + + ....
ç ÷
è 2! 3! ø
æ m 2 m 3 ö
+
m m e= + 2  m 1 m + + + .... = m m 2
+
ç ÷
è 2! 3! ø
2
Thus, Var(r) = m + m - m = m.
2
Also standard deviation s = m .
(c) The values of m , m , b and b
3 4 1 2
It can be shown that m = m and m = m + 3m .
2
3 4
m 2 3 m 2 1
 b = 3 = 3 =
1
m m m
2
Since m is a positive quantity, therefore, b is always positive and hence the Poisson
1
distribution is always positively skewed. We note that b  0 as m  , therefore the
1
distribution tends to become more and more symmetrical for large values of m.
m 4 m + 3m 2 1
Further, b = = = 3 +  3 as m  . This result shows that the
2 2 2
m m m
2
distribution becomes normal for large values of m.
(d) Mode
As in binomial distribution, a Poisson variate r will be mode if

P (r  ) 1  P ( ) r  ( P r + ) 1

P
The inequality (r  ) 1  P ( ) r can be written as

e  m .m r 1 e  m .m r m
 Þ 1 Þ r  m .... (1)
(r  ) 1 ! ! r r

Similarly, the inequality ( ) r  P (r + ) 1 can be shown to imply that
P
r  m - 1 .... (2)

Combining (1) and (2), we can write m - 1 £ r £ m.
Case I. When m is not an integer

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