where m is the observed number of arrivals in n unit time intervals. Similarly,
since represents the average number of arrivals in time interval [0,t).
Also it is seen from Equation (6.47) that, as expected, the variance, as well
as the mean, increases as the mean rate increases. The Poisson distribution for
several values of is shown in Figure 6.3. In general, if we examine the ratio of
and as we did for the binomial distribution, it shows that
in cr eases monotonically and then decreases monotonically as k
in cr eases, reaching its maximum when k is the largest integer not exceeding t.
Example 6.11.Problem: traffic load in the design of a pavement system is
an important consideration. Vehicles arrive at some point on the pavement in
0.60.40.2(^0) k k
k
pk(0,t)
pk(0,t)
pk(0,t)
0
0.2
0.4
(^024681012)
0
0.2
0.1
0.3
(a) (b)
(c)
012345 0123456
Figure6. 3 Poissondistribution forseveralvaluesof
SomeImportantDiscreteDistributions 177
t,
t
pk(0,t) pk 1 (0,t),
pk(0,t)
pk(0,t), t:(a)t 0 :5;(b)
t 1 :0;(c)t 4 : 0