Practice 5: Pure Birth, Pure Death, Generalized Poisson Model
April 28, 2009
Example 15.4, Exercise 1 on page 559.
PURE BIRTH
Babies are born at the rate of one birth every 12 minutes. The time between births
has exponential distribution. Find the following:
(a) The average number of births per year.
(b) The probability that
n
births will occur in any one day.
(c) The probability of issuing 50 birth certiﬁcates in 3 hours given that 40 certiﬁ
cates were issued during the ﬁrst 2 hours.
(d) Suppose that a clerk enters the information from birth certiﬁcates into a com
puter whenever 5 certiﬁcates have accumulated. What is the probability that
the clerk will be entering a new batch every hour?
Solution:
Recall, the number of arrivals (at rate
λ
) during a speciﬁed period of time
T
has the
following properties:
(1) The expected number of arrivals is
λT
.
(2) The probability of
n
arrivals (
n
≥
0 integer) in period
T
is
p
n
(
T
) =
(
λT
)
n
e

λT
n
!
.
First, we ﬁnd the birth rate per day:
λ
=
1
12
×
60
×
24 = 120 births per day
.
(a) The expected number of births per year is
λt
= 120
×
365 = 43
,
800 births per year
.
(b) This is
p
n
(
T
) where
T
is a day and
n
= 0, so we have
p
0
(1) =
(120
×
1)
0
e

120
×
1
0!
=
e

120
≈
0
.
1
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View Full Document(c) This is the same as the probability of issuing 10 certiﬁcates in an hour. The
rate is 1 certiﬁcate per 12 minutes or 5 certiﬁcates per hour. So the probability
is
p
n
(
T
) for
n
= 10 and
T
= 1 hour, i.e.,
p
10
(1) =
(5
×
1)
10
e

5
×
1
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 Spring '09
 Nedich
 Probability theory, Exponential distribution, average number

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