Fundamentals of Probability and Statistics for Engineers

as given by Equation (6.40), we have

Comparing this expression with Equation (7.59), we can establish the result
that the interarrival time between Poisson arrivals has an exponential distribu-
tion; the parameter in the distribution of T is the mean arrival rate associated
with Poisson arrivals.

Example 7.6.Problem: referring to Example 6.11 (page 177), determine the
probability that the headway (spacing measured in time) between arriving
vehicles is at least 2 minutes. Also, compute the mean headway.
Answer: in Example 6.11, the parameter was estimated to be 4.16 vehicles
per minute. Hence, if T is the headway in minutes, we have

The mean headway is

Since interarrival times for Poisson arrivals are independent, the time required
for a total of n Poisson arrivals is a sum of n independent and exponentially
distributed random variables. Let Tj, j 1, 2,... , n, be the interarrival time
between the (j 1)th and jth arrivals. The time required for a total of n arrivals,
denoted by Xn,is

where Tj,j 1,2,...,n, are in dependent and exponentially distributed with the
same parameter. In Example 4.16 (page 105), we showed that Xn has a
gamma distribution with 2 when n 2. The same procedure immediately
shows that, for general n, Xn is gamma-distributed with n. Thus, as stated,
the gamma distribution is appropriate for describing the time required for a
total of Poisson arrivals.

Example 7.7.Problem: ferries depart for trips across a river as soon as nine
vehicles are driven aboard. It is observed that vehicles arrive independently at
an average rate of 6 per hour. Determine the probability that the time between
trips will be less than 1 hour.
Answer: from our earlier discussion, the time between trips follows a gamma
distribution with 9 and 6. Hence, let X be the time between trips in

216 Fundamentals of Probability and Statistics for Engineers

P[X"0,t)ˆ0]ˆet

FT…t†ˆ

1 et; fort 0 ;

0 ; elsewhere:



… 7 : 62 †





P…T 2 †ˆ

Z 1

2

fT…t†dtˆ 1 FT… 2 †ˆe^2 …^4 :^16 †ˆ 0 : 00024 :

mTˆ

1



ˆ

1

4 : 16

minutesˆ 0 :24 minutes:

ˆ



XnˆT 1 ‡T 2 ‡‡Tn; … 7 : 63 †

ˆ



ˆˆ

ˆ



ˆ ˆ