Page 233 - Fundamentals of Probability and Statistics for Engineers

216 Fundamentals of Probability and Statistics for Engineers

P[X"0, t) ˆ 0] ˆ e t as given by Equation (6.40), we have
t
1 e ; for t 0;
F T t†ˆ 7:62†
0; elsewhere:

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

Z 1
P T 2†ˆ f t†dt ˆ 1 F T 2†ˆ e 2 4:16† ˆ 0:00024:
T
2
The mean headway is
1 1
m T ˆ ˆ minutes ˆ 0:24 minutes:
4:16
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 T j , j ˆ 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 X n , is
X n ˆ T 1 ‡ T 2 ‡ ‡ T n ; 7:63†

where T j , j ˆ 1, 2, .. . , n, are independent and exponentially distributed with the
same parameter . In Example 4.16 (page 105), we showed that X n has a
gamma distribution with ˆ 2 when n ˆ 2. The same procedure immediately
shows that, for general n, X n is gamma-distributed with ˆ 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 ˆ 9 and ˆ 6. Hence, let X be the time between trips in

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