Exercise 4.24: Compute probabilities with the Poisson
Suppose that over a period oft
time units, a particular uncertain event
happens (on average)νt
times. The probability that there will bex
such events in a time period t is approximately given by the formula
P(x,t, ν) =
This formula is known as the Poisson distribution. (It can be shown that
(4.9) arises from (4.8) when the probabilityp of experiencing the event in
asmall time intervalt/n isp =νt/n and we letn→∞ .) An important
assumption is that all events are independent of each other and that the
probability of experiencing an event does not change signiﬁcantly over
time. This is known as a Poisson process in probability theory.
a) Implement (4.9) in a a function Poisson(x, t, nu), and make a
program that readsx,t, andν from the command line and writes out
the probabilityP(x,t,ν). Use this program to solve the problems below.
average, 5 taxis pass this street every hour at this time of the night. What
is the probability of not getting a taxi after having waited 30 minutes?
Since we have 5 events in a time period oft
=ν = 5.
The sought probability is thenP(0,1/2, 5). Compute this number. What
is the probability of having to wait two hours for a taxi? If 8 people need
two taxis, that is the probability that two taxis arrive in a period of 20
c) In a certain location,10 earthquakes have beenrecorded during
the last 50 years. What is the probability of experiencing exactly three
earthquakes over a period of 10 years in this area? What is the probability
that a visitor for one week does not experience any earthquake? With
10 events over 50 years we haveνt
=ν·50years= 10events, which
impliesν = 1/5 event per year. The answer to the ﬁrst question of having
second question asks forx = 0 events in a time period of 1 week, i.e.,
t= 1/52 years, so the answer is P(0,1/52,1/5).
of the reports you write and that this number shows an average of six
misprints per page. What is the probability that a reader of a ﬁrst draft of
one of your reports reads six pages without hitting a misprint? Assuming
that the Poisson distribution can be applied to this problem, we have
as 1 page andν·1 = 6, i.e.,ν = 6 events (misprints) per page.
The probability of no events in a “period” of six pages is P(0, 6,6).