14
Homework 14 - Additional Problems
14.1. Let Xt be a branching process with birth rate and death rate (see Example 4.4 in
the book).
a) Let M2,n (t) = E [Xt2 | X0 = n]. Use the Kolmogorov forward equations to derive a
system of dierential equations for
13
Homework 13 - Additional Problems
13.1. In the standard branching process model individuals reproduce ospring at rate and
die at rate (see Example 4.4 in the book). Consider the following modication where there
the population can also grow by immigrati
13
Homework 13 - Additional Problems
13.1. In the standard branching process model individuals reproduce ospring at rate and
die at rate (see Example 4.4 in the book). Consider the following modication where there
the population can also grow by immigrati
12
Homework 12 - Additional Problems
12.1. Consider the continuous time Ehrenfest process Xt with N balls (that is, each ball
switches between the two urns at rate 1 independently of all the other balls).
a) Compute the jump rates q (i, j ) for this Marko
11
Homework 11 - Additional Problems
11.1. In this problem we will analyze the length of busy periods for an M/G/1 queue. That
is the arrivals are according to a Poisson process with rate , and the service times are
i.i.d. with some general distribution (
10
Homework 10 - Additional Problems
10.1. Consider the following modication to the Poisson janitor example (example 3.5 in the
book). Suppose that the average lifetime of the lightbulbs is F = 60 and that the janitor
comes to check on the bulb according
10
Homework 10 - Additional Problems
10.1. Consider the following modication to the Poisson janitor example (example 3.5 in the
book). Suppose that the average lifetime of the lightbulbs is F = 60 and that the janitor
comes to check on the bulb according
9
Homework 9 - Additional Problems
9.1. Let N be a non-homogeneous Poisson point process on (0, ) with density (x) = x,
and let N (t) = N (0, t]) be the number of points in the interval (0, t].
a) Compute E [N (5) | N (2) = 3].
b) Compute P (N (2) = 3 | N
8
Homework 8 - Additional Problems
8.1. Suppose that N = i1 (Xi ,Yi ) is a homogeneous Poisson point process on R2 with
intensity . Let R = mini1 Xi2 + Yi2 be the distance of the closest point to the origin.
a) Compute the density fR (t) of R.
Hint: rst c
7
Homework 7 - Additional Problems
7.1. Let X1 , X2 , . . . Xn be independent, identically distribution random variables all with
distribution Exp(). Let X = maxcfw_X1 , X2 , . . . , Xn .
a) Prove that X has the same distribution as
Yj Exp(j ) for j = 1,
6
Homework 6 - Additional Problems
6.1. Consider a random walk on the following graph.
1
2
3
4
6
5
7
9
8
10
If the random walk starts at the top of the triangle (state 1), what is the distribution when
it exits the bottom of the triangle. That is, let T =
5
Homework 5 - Additional Problems
5.1. (This is exercise 1.8 in Introduction to Stochastic Processes by Lawler).
Consider a simple random walk on the graph below. (Recall that simple random walk on a
graph is the markov chain which at each time moves to
5
Homework 5 - Additional Problems
5.1. (This is exercise 1.8 in Introduction to Stochastic Processes by Lawler).
Consider a simple random walk on the graph below. (Recall that simple random walk on a
graph is the markov chain which at each time moves to
4
Homework 4 - Additional Problems
4.1.
a) Let Xn be a Markov chain on I = cfw_1, 2, 3, 4 with transition probability matrix
1a
a
0
0
b
1b
0
0
, with 0 < a, b, c, d < 1.
p=
0
0
1c
c
0
0
d
1d
Compute limn pn (x, y ) for any x, y I .
Note that this limit e
3
Homework 3 - Additional Problems
3.1. There is a typo on page 20 of the book, just after the bold heading Using the TI83
calculator is easier. The book says
.we write (1.10) in matrix form as
.2 .1 1
(1 , 2 , 3 ) .2 .5 1 = (0, 0, 1)
.3
.3 1
If we let A
3
Homework 3 - Additional Problems
3.1. There is a typo on page 20 of the book, just after the bold heading Using the TI83
calculator is easier. The book says
.we write (1.10) in matrix form as
.2 .1 1
(1 , 2 , 3 ) .2 .5 1 = (0, 0, 1)
.3
.3 1
If we let A
2
Homework 2 - Additional Problems
2.1. Let cfw_Xn n0 be a Markov chain on a nite state space I . Dene Tcov to be the cover
time of the Markov chain to be the rst time that the Markov chain has visited every site
in I . That is,
Tcov = mincfw_n 0 : i I, k
1
Homework 1 - Additional Problems
1.1. Let cfw_Xn n0 be a random walk on Z with momentum that we discussed in class. That
is, let (1/2, 1) be xed and let
X0 = 0,
P (X1 = 1) = P (X1 = 1) = 1/2,
and
P (Xn+1 Xn = Xn Xn1 ) = ,
P (Xn+1 Xn = (Xn Xn1 ) = 1 .
Th