MASSACHUSETTS
INSTITUTE
OF
TECHNOLOGY
Fall 2007
6.436J/15.085J±
Final exam, 1:30–4:30pm, (180 mins/100 pts)
12/19/07±
Problem
1:
(24 points)
During the time interval
[0
,t
]
, men and women arrive according to independent Poisson
processes with parameters
λ
1
and
λ
2
, respectively. With the exception of part (e), just
provide answers (possibly based on your intuitive understanding)—justiﬁcations are not
required.
(a) (3 pts.) Let
[
a,b
]
be an interval contained in
[0
]
. Give a formula for the prob
ability that the total number of male arrivals during the interval
[
]
is equal
to 7.
(b) Out of all the people who arrived during
[0
]
, we select one at random, with each
one being equally likely to be selected.
(i) (3 pts.) Write an expression for the probability that the selected person is
male.
(ii) (3 pts.) Suppose that the randomly selected person tells us that he/she ar
rived at a particular time
τ
.
What is the conditional probability that this
person is male?
(iii) (3 pts.) Write an expression (as simple as you can) for the expected time at
which the selected person arrived.
(c) (4 pts.) Suppose that
0
< a < b < t
. Let
N
1
be the number of male arrivals
during
[0
,b
]
. Let
N
2
be the number of female arrivals during
[
a,t
]
. What is the
probability mass function of
N
1
+
N
2
?
(d) (4 pts.) Suppose that in (c) above we are told that
N
1
+
N
2
=
10
. Find the
conditional variance of
N
1
, given this information.
(e) (4 pts.) Find a good approximation for the probability of the event
{
the number of arriving men during
[0
]
is at least
λ
1
t
}
,
when
t
is large, and justify the approximation.
Solution:
(a)
e
−
λ
1
(
b
−
a
)
(
λ
1
(
b
−
a
))
7
7!
(b)(i)
λ
1
/
(
λ
1
+
λ
2
)
(ii) Since the time a person has arrived is independent of whether he was classiﬁed into
male or female, the answer is the same as in (i).
1
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View Full Document(iii) The distribution of a randomly selected arrival is uniform over
[0
,t
]
. Its expectation
is
t/
2
.
(c) Male and female arrivals are independent processes, and the sum of independent
Poisson random variables is again Poisson. The answer is Poisson with parameter
λ
1
b
+
λ
2
(
t
−
a
)
.
(d) Since each arrival
N
1
+
N
2
independently comes from
N
1
with probability
λ
1
b
p
=
,
λ
1
b
+
λ
2
(
t
−
a
)
the distribution of
N
1
conditional on
N
1
+
N
2
=
10
is binomial with parameters
n
=
10
and
p
. Its variance is
10
p
(1
−
p
)
.
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 Fall '08
 DavidGarnarnik
 Probability theory, Xn

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