ISEN 609
Quiz 3
Oral Quiz
Oral Quiz
There are 25 questions in total
are 25 questions in total
At the end of the quiz, based on your
answers score on 10 would be assigned
answers a score on 10 would be assigned
We will randomly select the person who get

Solutions to A Few More DTMC Modeling Exercises from Old Homeworks
1. At time n if the mouse is alive (i.e. Xn = 1) then at time n + 1 there are (N 1) (N 1) possible
arrangements, of which N 2 will result in the cat and the mouse being in the same node. T

Homework 4
1. There is an infinite supply of light bulbs, and Zi is the lifetime of the ith light bulb. cfw_Zi , i 1
is a sequence of iid discrete random variables with
P cfw_Zi = k = pk
where pk 0,
X
k = 1, 2, . . . ,
pk = 1. At time 0 the first light bu

Homework 1
1. A pair of dice is rolled until a sum of either 5 or 7 appears. Find the probability
that a 5 occurs first. (You are welcome to verify the solution using a simple
simulation)
Hint: Let En denote the event that a 5 occurs on the nth roll and n

Homework 3
1. Suppose the pdf of X is given by
(
f (x) =
1
xex/2 ,
4
0,
x>0
otherwise.
Calculate the LST. Using the LST calculate E(X) and also verify E(X) using the pdf
directly.
2. Bus A will arrive at a station at a random time uniformly distributed be

Solutions to homework 5
1.
a. Irreducible, aperiodic and positive recurrent. In this case P () exists and equals
P () =
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
b. Irreducible, periodic and positive recurrent. In th

Homework 2
1. Let the probability density function of X be given by
(
fX (x) =
c(4x 2x2 ), 0 < x < 2
0,
otherwise.
(a) What is the value of c?
(b) What is the cumulative distribution function of X?
(c) P cfw_ 21 < X < 32 = ?
2. The median of a continuous

Solutions to homework 4
1. Suppose Xn = j, i.e. the age of the light bulb that is on at time n is j Zi > j. Then
j + 1 if bulb doesnt fail Zi > j + 1
Xn+1 =
0
otherwise
Hence cfw_Xn , n 0 is a DTMC with transition probabilities
P
k=j+2
P cfw_Xn+1 = j + 1|

Extra problems on Topic 1 in ISEN 609
1. There are two mutually independent paths to go from location A to location B. The first path
takes a random time exponentially distributed with mean 10 minutes, while the second path
takes a random time uniformly d

Class Problems: DTMC
1. A company uses two forecasting tools to predict the demand of its product. Tool i is effective
w.p. pi (for i = 1, 2). If the nth prediction uses tool i and it is observed to be effective, then
the (n + 1)st prediction is also done

Class Problems
1. The following problem was posed and solved in the 18th century by Daniel Bernoulli.
Suppose that a jar contains 2N cards, two of them marked 1, two marked 2, two
marked 3, and so on. Draw out m cards at random. What is the expected numbe

Homework 1
1. A pair of dice is rolled until a sum of either 5 or 7 appears. Find the probability
that a 5 occurs rst. (You are welcome to verify the solution using a simple
simulation)
Hint: Let En denote the event that a 5 occurs on the nth roll and no