632 Introduction to Stochastic Processes Fall 2002
Midterm Exam II
Instructions: Hand in problem 1 for 50 points, problem 2 for 30 points,
and one other problem for 20 points. Show calculations and justify nonobvious statements for full credit.
1. Fix a c
632 Introduction to Stochastic Processes Fall 2002
Final Exam
Instructions: Hand in all four problems. The points add up to 100.
Show calculations and justify non-obvious statements for full credit.
1. (a) (15 pts) Suppose Xn is a discrete-time Markov cha
632 Introduction to Stochastic Processes Fall 2003
Midterm Exam II
Instructions: Justify non-obvious statements for full credit. Quote results from class accurately. Points add up 100.
1. (a) (20 pts) Let N be a rate Poisson process. Find the conditional
632 Introduction to Stochastic Processes Fall 2003
Final Exam
Instructions: Justify non-obvious statements for full credit. Quote results from class accurately. Points add up to 200.
1. Consider the discrete-time Markov chain with state space
S = Z+ = cfw
632 Introduction to Stochastic Processes Spring 2004
Midterm Exam I
Instructions: Show calculations and give concise justications for full
credit. Dont forget that theoretical ideas can help avoid tricky computations.
The points add up 100.
General notati
632 Introduction to Stochastic Processes Spring 2004
Midterm Exam II
Instructions: Show calculations and give concise justications for full
credit. Points add up 100.
1. (20 pts) Eileen is catching sh at the Poisson rate of per hour. Each
sh is a salmon w
632 Introduction to Stochastic Processes Spring 2004
Final Exam
Instructions: Show calculations and give concise justications for full
credit. Points add up to 200.
1. Four cages labeled A, B , C and D are connected by tubes as indicated
in the diagram be
Math 632 Fall 2011
Homework 0 Solutions
1. We toss a biased coin N times. The coin shows head with probability p (0, 1) and tail with
probability 1 p. Compute the probability that:
(a) all the tosses result in tails.
(b) the coin shows head at least once.
Math 632 Spring 2012
Homework 1 Solutions
1. Use generating functions to compute the third moment of a Poisson random variable with
parameter .
Solution: Let X be a Poisson() random variable then PX (s) = e(s1) and
PX (1) = 3 = EX (X 1)(X 2) = EX 3 3EX 2
Math 632 Fall 2011
Homework 2 Solutions
1. 2.2, page 147
If the original Markov chain had transition probabilities pij then the new one will have
p(i,j ),(k,l) = P (Xn+1 = k, Yn+1 = l|Xn = i, Yn = j ) =
P (Xn+1 = k, Yn+1 = l, Xn = i, Yn = j )
P (Xn = i, Y
Math 632 Spring 2012
Homework 3 Solutions
1. 2.16 a), p. 151
One can do this with the methods learned about absorption, but its a lot easier than that.
Note that if we are at state 1 or 2 then with probability 1/2 we go to 0 and with probability
1/2 we go
Math 632 Spring 2012
Homework 4 Solutions
1. 3.4, p. 282
Although the problem does not state this, we can assume that pk = P (N = k ). Then the
distribution function of SN = X1 + + XN is
P (SN x) =
P (SN x, N = k ) =
k=0
pk F k (x).
P (Sk x)P (N = k ) =
k
Math 632 Spring 2012
Homework 5 - Solutions
1. 4.3, page 350
For x 1, let I (x) satisfy
I (x)
1
1
dt = x ln(I (x) = x I (x) = ex .
t
Then, by the work in section 4.3, we know that if N is a unit rate homogeneous Poisson
process with points n , then for t
Math 632 Spring 2012
Homework 6 - Solutions
1. 5.11, page 453
Dierentiating gives
P (t) =
2e5t 2e5t
3e5t 3e5t
2 2
. Thus if P (t) is a transition probability matrix of a CTMC
3 3
then the generator must be equal to A. But P (0) is the identity matrix and
632 Introduction to Stochastic Processes Spring 2007
Midterm Exam 1
Instructions: Show calculations and give concise justications for full
credit. Points add up to 100.
In-Class Part
1. Consider the Markov chain
transition matrix
1/3
1/2
P=
1/4
0
on the s
632 Introduction to Stochastic Processes Spring 2007
Final Exam
Instructions: Show calculations and give concise justications for
full credit. Points add up to 200.
1. Born again branching process. Let cfw_pk 0k< be the ospring
distribution of a branching
Probabilities and Random Variables
This is an elementary overview of the basic concepts of probability theory.
1
The Probability Space
The purpose of probability theory is to model random experiments so that we can draw inferences
about them. The fundamen