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
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 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 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 5
Due: April 19, beginning of the class. Late homework will not be accepted.
1. 4.3, page 350
2. 4.5, page 350
3. 4.15, page 354
4. 4.22 a), page 356 (You only need to check the conditions for intervals.)
5. 4.32, page 359
6.
Math 632 Spring 2012
Homework 6
Due: May 8, beginning of the class. Late homework will not be accepted.
1. 5.11, page 453
2. 5.15, page 454 (Do not do the part about binary splitting.)
Assume that if the process dies out (i.e. X (t) = 0) then a new partic
Math 632 Spring 2012
Homework 4
Due: March 29, beginning of the class. Late homework will not be accepted.
1. 3.4, p. 282 The answer will be in terms of the generating function of pk and the Laplace
transform of F .
2. A certain component of a complicated
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
Due: March 13, beginning of the class. Late homework will not be accepted.
1. 2.16 a), p. 151
2. 2.21, p. 154
Hints: Show that P n = P and that pii > 0 for all i S .
3. 2.27, p. 155
4. 2.48, p. 161
5. 2.52, p. 161
6. (From
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 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 Spring 2012
Homework 2
Due: February 23, beginning of the class. Late homework will not be accepted.
1. 2.2, page 147
Hint: you will need the transition probabilities for jumps of the form (i, j ) (k, l).
2. 2.4, page 148
3. Consider a Markov cha
Math 632 Spring 2012
Homework 1
Due: February 9, beginning of the class. Late homework will not be accepted.
1. Use generating functions to compute the third moment of a Poisson random variable with
parameter .
2. 1.3, page 52
3. 1.5 b), page 53 (Hint: th
Math 632 Spring 2012
Homework 0
Due: January 26, beginning of the class. Late homework will not be accepted.
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) a