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Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2002)

Final Exam Announcements

Exam: (closed-book; three handwritten 2-sided 8.5x11 formula sheets and calculator permitted.

Time: 3 Hours

Content: All topics discussed in
Lectures up to and including Lecture 25
Textbook chapters 1-7
Recitations 1-14, Tutorials 1-14 and Problem Sets 1-10

Marathon Oﬃce Hours: The TAs will jointly hold oﬃce hours at selected times twice a day
before the exam. Check the course webpage on the last day of Lectures.

Optional Review Session: The TAs will provide a two-hour review session during which the
review problems below will be solved. We suggest working through these problems before attending
the review session. For students unable to attend, solutions will be posted on the course web page
immediately after the review session. Details for the review session are:

Time: 2 hour session

Also attached is a practice ﬁnal exam from the previous semester and its solution. Working through
the exam before looking at the solutions is a good way to identify areas meriting additional study.

Review Problems

1.	 Joe Lucky remembers to play the lottery on any given week with probability p, independently
of whether he played on any other week. Each time he plays, he has a probability q of winning,
again independently of everything else. During a ﬁxed time period of n weeks, let X be the
number of weeks that he played the lottery and Y the number of weeks that he won.

(a)	 What is the probability that he played the lottery on any particular week, given that he
did not win anything on that week?
(b) Find the conditional PMF pY |X (y | x).
(c) Find the joint PMF pX,Y (x, y).
(d)	 Find the marginal PMF pY (y). Hint: One possibility is to start with the answer to part
(c), but the algebra can be messy. But if you think intuitively about the procedure that
generates Y , you may be able to guess the answer.
(e) Find the conditional PMF pX|Y (x | y). Do this algebraically using previous answers.
(f)	 Rederive the answer to part (e) by thinking as follows. For each one of the n − Y weeks
that he did not win, the answer to part (a) should tell you something.

Page 1 of 8

Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2002)

In all parts of this problem, make sure to indicate the range of values to which your PMF
formula applies.

2.	 A 6.041 graduate opens a new casino in Las Vegas and decides to make the games more
challenging from a probabilistic point of view. In a new version of roulette, each contestant
spins the following kind of roulette wheel. The wheel has radius r and its perimeter is divided
into 20 intervals, alternating red and black. The red intervals (along the perimeter) are twice
the width of the black intervals (also along the perimeter). The red intervals all have the same
length and the black intervals all have the same length. After the wheel is spun, the center
of the ball is equally likely to settle in any position on the edge of the wheel; in other words,
the angle of the ﬁnal ball position (marked at the ball’s center) along the wheel’s perimeter
is distributed uniformly between 0 and 2π radians.

(a) What is the probability that the center of the ball settles in a red interval?
(b)	 Let B denote the event that the center of the ball settles in a black interval. Find the
conditional PDF fZ|B (z), where Z is the distance, along the perimeter of the roulette
wheel, between the center of the ball and the edge of the interval immediately clockwise
from the center of the ball?
(c) What is the unconditional PDF fZ (z)?

Another attraction of the casino is the Gaussian slot machine. In this game, the machine
produces independent identically distributed (IID) numbers X1 , X2 , ... that have normal dis­
tribution N (0, σ 2 ). For every i, when the number Xi is positive, the player receives from the
casino a sum of money equal to Xi . When Xi is negative, the player pays the casino a sum
of money equal to |Xi |.

(d)	 What is the standard deviation of the net total gain of a player after n plays of the
Gaussian slot machine?
(e) What is the probability that the absolute value of the net total gain after n plays is
√
greater than 2
nσ?

3.	 Let a continuous random variable X be uniformly distributed over the interval [−1, 1]. Derive
the PDF fY (y) where:

(a) Y = sin( π X)
2
(b) Y = sin(2πX)

4.	 People who wish to use a particular mailbox arrive at the box in a Poisson manner with
average arrival rate of λ customers per hour. Independently, each user of the mailbox wishes
2
to mail either one letter (with probability 3 ) or one parcel (with probability 1 ). Although the
3
users arrive one at a time in a Poisson manner, the ith user is accompanied by N i non-user
friends, where the Ni ’s are independent and identically-distributed random variables with an
associated transform
1
1
1 2s
MN (s) = +
es + e
.

2
3
6

Page 2 of 8

Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2002)

(a)	 Determine the expected value of Y , the total number of people arriving at the mailbox
during a three hour interval.
(b)	 At 3 P.M. today, we shall begin counting letters and parcels as they are brought to the
mailbox. What is the probability we shall see a total of exactly three parcels by the time
the ﬁfth letter arrives?
(c) Determine the probability that exactly three out of the next eight users will mail parcels.
(d)	 If we make an equally likely selection from all people (users and non-users) who arrive
at the mailbox in the last week, what is the probability we select a person who was
accompanied when he or she arrived at the mailbox
(e)	 Determine either the PMF or the transform for K, the total number of people arriving
at the mailbox during a particular hour.
(f)	 For a time selected by random incidence, determine the transform for T , the total time
interval from the arrival of the third previous user until the arrival of the ﬁfth future
user of the mailbox.

5.	 Mary loves gambling. She starts out with \$200. She can bet either \$100 or \$200 (assuming
she has suﬃcient funds), and wins with probability p.

(a)	 Assuming that she stops when she runs out of money or when she has reached \$400,
what is the optimal betting strategy? (i.e., how much should she bet when she has \$100,
\$200, \$300? The amount she bets does not have to be the same amount at each time.)
(b)	 What is the expected number of transitions until she either runs out of money or reaches
\$400 for p = 0.75 under the optimal strategy?

6.	 Wombats and dingos arrive in a Poisson manner to a particular water hole in the Australian
Outback. The arrival rates of wombats and dingos are 2 and 4 per hour, respectively. Each
animal will stay and drink until the next animal arrives to take over. No other animals visit
the water hole.

(a)	 What is the expected number of animals (wombats or dingos) that visit the water hole
in a 24-hour period?
(b)	 Given that a wombat is currently drinking, what is the probability that the next animal
to visit is a dingo?

Crocodile Dundee arrives at the water hole at a random time and leaves the water hole
immediately after the 900th animal he sees departing the water hole.

(c) How long, on average, will Crocodile Dundee have to wait to see a dingo?
(d)	 Consider the ﬁrst dingo that Crocodile Dundee sees. How long, on average, does this
dingo spend at the water hole?
(e)	 What does Chebyshev’s inequality tell you about the probability that Crocodile Dundee
stays at the water hole for between 140 and 160 hours?
(f)	 Using the Central Limit Theorem, what is the probability that Crocodile Dundee stays
at the water hole for between 140 and 160 hours?

Page 3 of 8

Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2002)

Spring 2002 Final Exam

Problem 0. (2 points) Write your TA’s name on the front of the booklet.

Problem 1. (46 points)
The student center lobby has two ATMs, one for cash-only transactions, and one for all
transactions.

λC                  p
Cash
1-p
All
λN
Customers (type-C) with cash-only transactions arrive according to a Poisson process with
rate λC per hour. Each type-C customer chooses the cash-only ATM with probability p, and
the all-transaction ATM with probability (1 − p), independently from other customers.
Customers (type-N) with non-cash transactions arrive according to an independent Poisson
process with rate λN per hour, and choose the all-transaction ATM.

(a)	 (5 points) Give a formula for the mean and variance of the number of customers with
cash-only transactions that arrive between 9 a.m. and 11 a.m.
(b)	 (5 points)A customer just arrived at the all-transaction ATM. Whatt is the probability
that he is a type-C customer?
(c)	 (5 points) Given that there was exactly one type-C customer between 9 a.m. and 10
a.m., ﬁnd the PDF of the arrival time of that customer.
(d)	 (5 points) Given that there was at least one type-C customer between 9 a.m. and 10
a.m., ﬁnd the PDF of the arrival time of the ﬁrst type-C customer.
(e)	 (6 points) Find the PDF of the square of the time elapsed between the arrival of the
ﬁrst and the second type-C customer.
(f)	 (7 points) Give a formula for the probability that the number of type-C customers that
arrive between 9 a.m. and 12 p.m. is at least twice the number of type-C customers
that arrive between 9 a.m. and 10 a.m. (If your formula involves a complicated sum or
integral, you do not need to evaluate it.)
(g)	 (6 points) Suppose that λC = 2 and that p = 0.98. Find a good numerical approximation
for the probability that out of the ﬁrst 100 type-C customers, exactly 95 choose the cash-
only ATM.
(h)	 (7 points) Give a formula for the expected time until at least one customer of each type
(that is, type-C and type-N) has arrived.

Page 4 of 8

Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2002)

Problem 2. (52 points)

Consider the discrete-time Markov chain described by the following transition diagram:

1/2           1/3

1              2

1/2         1/3
1/3
1           2/3                           1/3
5            6            7                  3            4

1/3         1                               1
2/3

(a)	 (5 points) Assume that the process starts at state 1, i.e. X0 = 1. Find the probability
that X2 = 5.
(b)	 (5 points) Assume that the process starts at state 1, i.e. X0 = 1. Find the probability
that X2 = 1 given that X3 = 6.
(c) (5 points) Find approximately the probability P(Xn = 3 | X0 = 3), when n is large.
(d) (5 points) Find approximately the probability P(Xn = 3 | X0 = 1), when n is large.
(e)	 (5 points) Find the expected time until the state leaves the set {1, 2}, given that the
process starts at state 1.
(f)	 (7 points) Given that the process starts at state 3, ﬁnd the expected value and the
variance of the time elapsed until state 4 is entered for the third time.
(g)	 (6 points) Suppose that the process starts at state 6. Find a good approximation for the
probability that state 5 is visited at least 30 times in the ﬁrst 144 transitions.
(h)	 (6 points) Suppose that the process starts at state 6. Let Vn be the number of visits to
state 7 in the ﬁrst n transitions. For each one of the expressions below, state whether
it converges in probability, and also whether it converges with probability 1. Whenever
the answer is yes, also state what the limit is.
Vn −E[Vn ]
i.	      √
n
Vn −E[Vn ]
ii.	       n
(i)	 (8 points) Suppose that the process starts at time 0 at state 3. Each time (including
time 0) that state 3 is visited, you win a prize, with probability 1/2. Each time that
state 4 is visited, you win a prize, with probability 1/4. Given the current state, the
event of winning a prize is assumed independent of everything else. Find the expected
value of the time at which you win a prize for the ﬁrst time.

Page 5 of 8

Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2002)

Spring 2002 Final Exam Solutions

Problem 0: (2 pts) Answers may vary.
Problem 1:

(a)	 (5 pts) The number of type-C customers that arrive between 9am and 11am is Poisson
with parameter 2λC , which has mean 2λC and variance 2λC .
(b) (5 pts) At the all-transaction ATM, type-C customers arrive with rate (1 − p)λ C and
type-N customers arrive with rate λN . So a customer at the all-transaction ATM is
(1−p)λC
type-C with probability (1−p)λC +λN .
(c)	 (5 pts) Let X be the time elapsed between 9am and the arrival time of the ﬁrst type-C
customer. Given that there was exactly one type-C customer between 9am and 10am,
that customer’s arrival time is uniformly distributed between 9am and 10am. Therefore,
X is uniformly distributed on the interval [0,1].
(d)	 (5 pts) Given that there was at least one type-C customer between 9am and 10am, we
know that the ﬁrst type-C customer arrived before 10am. So we want to ﬁnd the PDF
of X given that X ≤ 1. For 0 ≤ x ≤ 1, we have

fX (x)    λC e−λC x
fX|X≤1 (x) =            =           .
P(X ≤ 1)   1 − e−λC

(e)	 (6 pts) Let Y be the time elapsed between the arrival of the ﬁrst and the second type-C
customer, so Y is exponentially distributed with rate λC . We may ﬁnd the PDF of
Z = Y 2 by ﬁnding the CDF of Z ﬁrst (notice that Z = Y 2 is monotonic because Y is
nonnegative). For z ≥ 0, we have
√             √
FZ (z) = P(Z ≤ z) = P(Y 2 ≤ z) = P(Y ≤        z) = 1 − e−λC z

d           d           √      λC    √
fZ (z) =      FZ (z) =    (1 − e−λC z ) = √ e−λC z
dz          dz                2 z
(f)	 (7 pts) Let A be the event that the number of type-C customers that arrive between
10am and 12pm is greater than or equal to the number of type-C customers that arrive
between 9am and 10am.
∞
�
P(A) =           P(A|k arrivals between 9am and 10am)P(k arrivals between 9am and 10am)
k=0
�∞
=         P(at least k arrivals between 10am and 12pm)P(k arrivals between 9am and 10am)
k=0
∞ ∞
��      (2λC )j e−2λC λk e−λC
C
=
k=0 j=k
j!         k!

(g)	 (6 pts) Let S100 be binomial with 100 trials and probability of success .02 (we need
the probability of success to be small in order to use the Poisson approximation of the

Page 6 of 8

Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2002)

binomial, so in this case a success represents choosing the all-transaction ATM). We may
approximate S100 as a Poisson random variable with parameter np = 100(.02) = 2.

P(out of 100 type-C customers, exactly 95 choose the cash-only ATM) = P(S 100 = 5)
25 e−2
≈
5!

(h)	 (7 pts) Let T1 be the time until the ﬁrst customer arrives, so T1 is exponentially dis­
tributed with rate λC + λN . Let T2 be the time elapsed between the ﬁrst customer
arrival and the ﬁrst arrival of a customer of opposite type, so the time until at least one
customer of each type has arrived is T1 + T2 .

E[T1 + T2 ] = E[T1 ] + E[T2 ]
= E[T1 ] + E[T2 |ﬁrst arrival is type-C]P(ﬁrst arrival is type-C)
+E[T2 |ﬁrst arrival is type-N]P(ﬁrst arrival is type-N)
1         1      λC        1   λN
=          +                +
λC + λ N    λN λC + λ N      λC λC + λ N

Problem 2:

(a) (5 pts) The only way to be at state 5 after two transitions, if we are currently at state
11
1, is 1 → 6 → 5. So P(X2 = 5|X0 = 1) = 2 3 .
(b)	 (5 pts) There are three ways to be at state 6 after 3 transitions, if we are currently at
state 1: 1 → 2 → 1 → 6, 1 → 6 → 5 → 6, and 1 → 6 → 7 → 6.
111
P(X2 = 1, X3 = 6|X0 = 1)                 232
P(X2 = 1|X3 = 6, X0 = 1) =                                =     111       11         12
P(X3 = 6|X0 = 1)            232   +   2 31   +   2 31

(c)	 (5 pts) For n is large, P(Xn = 3|X0 = 3) = π3 . Noticing states 3 and 4 form a birth-
death chain, we solve the following set of equations to ﬁnd the steady-state probability
of being at state 3:
1
π3 = π 4
3
π3 + π 4 = 1
3
which gives us π3 = 4 .
(d)	 (5 pts) For n is large, P(Xn = 3|X0 = 1) = a13 π3 , where a13 is the probability of being
absorbed by state 3’s recurrent class if we are currently at state 1. We solve the following
set of equations to ﬁnd a13 :
1
a13 =     a23
2
1      1      1
a23 =     a23 + a13 +
3      3      3
13     1
which gives us a13 π3 =   34   = 4.

Page 7 of 8

Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2002)

(e)	 (5 pts) We solve the following set of equations to ﬁnd the expected time until absorption
from state 1.
1

µ1 = 1 + µ2

2
1    1

µ2 = 1 + µ1 + µ2

3    3
7

which gives us µ1 = 3 .

(f) (7 pts) Given we are currently at state 3, the time until we enter state 4 is geometric
1

with parameter p = 3 . Given we are currently at state 4, the time until we enter state 3

is 1. So the time until we enter state 4 for the third time is the sum of three independent

−
geometric random variables plus 2, which has mean 3 p + 2 = 11 and variance 3 1p2p = 18.

1

(g) (6 pts) Suppose the process starts at state 6. Then at all even times we are at state
1

6.	 At all odds times we are at state 5 with probability 3 or at state 7 with probability

2

3 , independently of where we were two transitions earlier. So the number of visits to
state 5 in the ﬁrst 144 transitions is a binomial random variable with 144 = 72 trials
2

1

and probability of success 3 . Using the Central Limit Theorem with a half-correction,
we have:
                               
1

S72 − E[S72 ]  29.5 − 72 3
P(S72   ≥ 30) = P                ≥ �              ≈ P(Z ≥ 1.375) = 1−Φ(1.375) = .0838.

σS72              1 2

72	2 3

(h) (6 pts) Vn is a binomial random variable with � n � trials and probability of success 2 .

2                                     3
Because a binomial random variable is the sum of independent, identically distributed

random variables, we know that Vn −E[Vn ] approaches a normal distribution as n → ∞

√
n
(by the Central Limit Theorem), and therefore does not converge, and that Vn −E[Vn ]
n
converges to 0 in probability and with probability 1 (by the Weak and Strong Laws of
Large Numbers respectively).
(i) (8 pts) Let hi be the expected time until we win a prize given we are currently at state
i. Using the total expectation theorem conditioned on whether we win or lose and on
where the chain transitions to, we set up the following set of equations:
1         1 2              1

�                         �
h3 =     (0) +         (1 + h3 ) + (1 + h4 )

2         2
3             3
1         3

h4 =     (0) +     (1 + h3 )

4         4
15

which gives us h3 =   13 .

Page 8 of 8

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