ORF 309
Solutions to Homework 1
Fall 2013
Due on Sep. 18, 2013
Exercise 1.4
Let E , F , G be three events.
a) only F occurs: F E c Gc
b) both E and F but not G: E F Gc
c) at least one event occurs: E F G
d) at least two events occur: (E F Gc ) (E F c G) (

ORF 309
Solutions to Homework 8
Fall 2013
Due on Nov. 20, 2013
Exercise 5.60
Let us use minutes as the units of time. Denote by N the given Poisson process with rate
. The event that two customers arrived during the rst hour is cfw_N60 = 2.
(a) The event

ORF 309
Solutions to Homework 10
Fall 2013
Due on Dec. 2, 2013
Exercise 3.9
To show X is a Gaussian process, we need to show that for any n 1, t1 , , tn [0, 1],
(Xt1 , , Xtn ) is a Gaussian vector.
It is equivalent to show for any Rn , n=1 i Xti is a Gaus

ORF 309
Solutions to Homework 9
Fall 2013
Due on Nov. 26, 2013
Exercise 1.13
We are given that X gamma(, ) (or X (, ). That is,
Pcfw_X dx =
ex (x)1
dx,
()
x0
so using the dx notation, we can write
Pcfw_cX dx = Pcfw_cX [x, x + dx]
x x dx
,+
cc
c
e(x/c) (x/

ORF 309
Solutions to Homework 11
Fall 2013
Not to be submitted
Exercise 4.2
Similarly to Example 4.4 on page 193-194, the system can be analyzed by using a Markov
chain with 8 states:
state
state
state
state
state
state
state
state
0
1
2
3
4
5
6
7
Today
R

ORF 309
Solutions to Homework 7
Fall 2013
Due on Nov. 13, 2013
Exercise 5.4
For each i cfw_A, B, C , dene
Ti := the time it takes for the clerk to serve customer i.
In order for A to still be in the post oce after the other two have left, B must nish befo

ORF 309
Solutions to Homework 2
Fall 2013
Due on Sep. 25, 2013
Exercise 2.16
Let X be the random variable describing the number of the people showing up for the specic
ight. As the airline believes that there is a 95 percent probability that someone who m

ORF 309
Solutions to Homework 3
Fall 2013
Due on Oct. 2, 2013
Exercise 2.40
As usual, we model the sequence of games as a sequence of Bernoulli trials, in other words,
as a Bernoulli process. Let
1 if A wins the ith game
Xi =
, i cfw_1, 2, ..
0 if B wins

ORF 309
Solutions to Homework 4
Fall 2013
Due on Oct. 9, 2013
Exercise 9.4
(a) (NEW QUESTION COMPARED TO BOOK) The system corresponding to the structure function (x) = x1 max(x2 , x3 , x4 )x5 is given by the series of the following three
subsystems:
comp

ORF 309
Solutions to Homework 5
Fall 2013
Due on Oct. 16, 2013
Exercise 3.3
By the denition of the conditional expectation and conditional probability,
3
E[X |Y = i] =
3
j Pcfw_X = j |Y = i =
j =1
j
j =1
Pcfw_X = j, Y = i
.
Pcfw_ Y = i
Expand Pcfw_Y = i,

ORF 309
Solutions to Homework 6
Fall 2013
Due on Nov. 6, 2013
Exercise 1
In this question, we will use Theorem 7 in Prof. Cnlars lecture notes on Branching Pro
cesses, which states that if 1, then = 1, while if > 1, then 0 < < 1 and is the
smallest soluti