Fundamentals of Probability and Statistics for Engineers

a random manner both in space (amplitude and velocity) and in time (arrival
rate). Considering the time aspect alone, observations are made at 30-second
intervals as shown in Table 6.1.
Suppose that the rate of 10 vehicles per minute is the level of critical traffic
load. Determine the probability that this critical level is reached or exceeded.
Let X(0,t) be the number of vehicles per minute passing so me point on the
pavement. It can be assumed that all conditions for a Poisson distribution are
satisfied in this case. The pmf of X(0, t) is thus given by Equation (6.44). From
the data, the average number of vehicles per 30 seconds is

H ence, an estimate of is 2.08(2) The desired probability is, then,

The calculations involved in Example 6.11 are tedious. Because of its wide
applicability, the Poisson distribution for different values of is tabulated
in the literature. Table A.2 in Appendix A gives its mass function for values
of ranging from 0.1 to 10. Figure 6.4 is also convenient for determining

Table 6. 1 Observedfrequencies(number of

observations)of 0 , 1 , 2 ,...vehiclesarrivingina

30-secondinterval(forExample6.11)

No.ofvehiclesper 30 s Frequency

018

132

228

320

413

57

60

71

81

90

Total 120

178 FundamentalsofProbabilityandStatisticsforEngineers



0 … 18 †‡ 1 … 32 †‡ 2 … 28 †‡‡ 9 … 0 †

120

 2 : 08 :

t ˆ 4 :16.

P‰X… 0 ;t† 10 Šˆ

X^1

kˆ 10

pk… 0 ;t†ˆ 1 

X^9

kˆ 0

pk… 0 ;t†

ˆ 1 

X^9

kˆ 0

… 4 : 16 †ke^4 :^16

k!

 1  0 : 992 ˆ 0 : 008 :

t

t