Homework 01
1. Suppose that
F ( x) =
0 if x 0
16 13 12
x + x + x if x [0, 1]
6
3
2
1 if x 1
(a) Convince me that F is a distribution function.
This function is clearly non-decreasing, continuous (and therefore, continuous from
above), and converges to 1 a
The transform method
The goal is that given a sequence of independent, idenctically distributed random variables with
a given distribution can be produce a random variable with another distribution? Sometimes
we can. For example, if we are given a sequenc
Markov Chains in Continuous Time
1. The Pure Death Process. Suppose that we have a population of M individuals. Suppose
that the lifetimes of these individuals are independent identically distributed random
variables whose common distribution function we
Homework 06 Solutions
Page 359, Number 36: Recall that for any complex number z ,
(z ) = E z N (t)
(t)n
exp(t)
n!
n=0
(tz )n
=
exp(t)
n!
n=0
= exp(t(z 1)
zn
=
Observe that S (t) 0 so for any non-negative real number p,
N (t)
p
S (t) = s
p
p
Xk
k=1
N (t)
Homework 02
1. I have 3 green balls and 3 red balls, and one white box and one black box. There are
three balls in each box. Once a minute I choose a ball at random from each box and
put it in the other box. Explain why this can be modeled as a Markov cha
Homework 03
1. Problem 13. Let Q = P r and let R = P nr where n r. Note that all the elements of
Q are positive by assumption, and each row of R has a positive element since the rows
of R each sum to 1. So for each q there is some p so that Rq,p > 0. Then
The rejection method
The goal is that given a sequence of independent, idenctically distributed random variables with
a given distribution can be produce a random variable with another distribution? Sometimes
we can. For example, if we are given a sequenc
Homework 05 Solutions
1. Suppose that Tk , k = 0, 1, 2, . . . are independent and identically distributed exponential
random variables, and we observe
T1
T2
T3
T4
T5
=
=
=
=
=
2
1
1/2
3
4
Graph the Poisson process N (t) for which these are the interarrval
Homework 04
1. First let us establish the hint, which is really an instance of a general fact. Suppose that
A, B and C are events and Pr(B C ) > 0. Then Pr(C ) Pr(B C ) > 0 and
Pr(A B C )
Pr(B C )
Pr(A B C )/ Pr(C )
=
Pr(B C )/ Pr(C )
Pr(A B |C )
=
Pr(B |
1. Consider the function F : (, ) [0, 1] given by
F ( x) =
2
3
sin
x
2
+
0 if x 0
if 0 x 1
1 if x 1
1
x
3
(a) (5 points) Verify that F is a probability distribution function. First of all notice
that F is well-dened, as it could be that it is dened two wa
The transform method
The goal is that given a sequence of independent, idenctically distributed random variables with
a given distribution can be produce a random variable with another distribution? Sometimes
we can. For example, if we are given a sequenc
1. Consider the function F : (, ) [0, 1] given by
F ( x) =
2
3
sin
x
2
+
0 if x 0
if 0 x 1
1 if x 1
1
x
3
(a) (5 points) Verify that F is a probability distribution function.
(b) (10 points) Describe explicitly how to simulate an observation from F using