Fall 2014
STAT 334
Zhu, M
University of Waterloo
Statistics 334
Problem Set 2
Due: Friday, October 24, 2014
1. Let X and Y be independent exponential random variables with rate parameter , i.e.,
fX (x
Stat 334
Midterm
21st June 2011
Name(Please Print):
Student ID:
Instructions
1. You have eighty minutes to complete the test.
2. If you need more working room, use the back of the preceding page, indi
Lecture 5
21st May 2013
Example: An insurance company classifies
policy holders as class A, B, C or D. The
probabilities of a randomly selected policy
holder being in these categories are:
Outcome
Pr
Lecture 2
9th May 2013
Example : Suppose in a Poisson Process,
there are 3 events on average per hour.
a)What is the probability of exactly 2 events
in the 1st hour?
b)What is the probability of at l
Lecture 4
16th May 2013
Steps to find the pdf of Y=h(X), given the
pdf of X:
1.Find the support of Y using a sketch
2.Find the cdf of Y, P(Yy), by writing the
event Yy in terms of X and y
3.Use the k
Lecture 6
23rd May 2013
Example: Suppose there are 25 different
types of coupons and suppose that each
time one obtains a coupon, it is equally
likely to be any one of the 25 types.
Compute the expec
Lecture 7
28th May 2013
When we are interested in finding the
distribution of a sum or linear combination
of independent random variables, the
multivariate MGF function is a very useful
tool.
Can yo
Lecture 8
30th May 2013
Approaches:
1.When we only have, for example two
random variables, and a single function
then the cdf of U can be found through
integration or summation and through
differenti
Lecture 3
14th May 2013
Later on we will look at Brownian motion,
which is used to describe the behaviour of
these small changes of an asset price over
time.
We will see that the change in value over
Stat 334
Midterm
21st June 2011
Name(Please Print):
Student ID:
Instructions
1. You have eighty minutes to complete the test.
2. If you need more working room, use the back of the preceding page, indi
Fall 2016
STAT 334
Zhu, M
University of Waterloo
STAT 334
Assignment #4
( Distributed: Tuesday, November 22, 2016; Due: Friday, December 2, 2016 )
1. Let X be an exponential random variable with F (x)
'
$
Example 1
Definition A state j is accessible from state i if there exists
n > 0 such that Pijn > 0.
Question The following facts are quite obvious indeed, but
how to formally prove them?
(a) If Pi
'
$
Example 1
For any given day, let X be the number of customers that visit
your shop (or website, .); and let Y be the number of sales you
make. Suppose
X Poisson()
and
Y |X Binomial(X, p).
What is
'
$
Example 1
Let (X, Y, Z) trinomial(n; p1 , p2 , p3 ), where p1 + p2 + p3 = 1,
X + Y + Z = n, and the joint distribution of (X, Y ) is
f (x, y) = P(X = x, Y = y)
=
n!
px1 py2 (1 p1 p2 )nxy .
x!y!(n
'
$
Example 1
Consider a Markov chain, cfw_Xn ; n = 0, 1, 2, ., with two states,
cfw_happy; sad,
and the following transition matrix:
2/3 1/3
.
P=
1/3 2/3
Suppose we are initially in the happy state,
'
$
Example 1
Consider a Markov chain with states cfw_A, B, C and transition
matrix:
1 0 0
.
0 0 1
(a) What is the limiting behavior of this Markov chain?
(b) If X0 =B, how many steps will it take on
'
$
Example 1
Consider a Markov chain with transition matrix:
1/3 0 1/3 1/3
1 0
0
0
.
0
1
0
0
0 1
0
0
(a) For each state, is it recurrent, transient, and/or absorbing?
(b) What is the limiting behavi
'
$
Question 1
The number of insurance claims over a certain fixed time period in
the future (e.g., next week, next month) is clearly a random
variable.
(a) Why is the Poisson distribution a reasonabl
STAT 334, Assignment #1, Solution Sketch
1. (a) By independence,
1
1
rT
E
e
=E
E erT ,
X +1
X +1
where
1
E
X +1
=
X
=
e x
1
x+1
x!
x=0
e
X
e
X
y1
x=0
y=x+1
=
x
(x + 1)!
y=1
=
y!
0
e X y
y!
0!
y=0
=
=
'
$
Simple Random Walk (SRW) on Z
A Markov chain, cfw_Xn ; n = 0, 1, 2, ., with state space
Z = cfw_., 3, 2, 1, 0, + 1, + 2, + 3, .
and transition probabilities
Pij = P(Xn+1
p,
= j|Xn = i) = 1 p,
0,
j
Lecture 18
18th July 2013
Recall:
Example:
For a standard Brownian motion Z(t), find
the probability that the first time the
process hits 1 happens after t=2.
W(t)>0, this is clear when you think o
Stat 333: Test 1 - Winter 2012
Department of Statistics and Actuarial Science, University of Waterloo
Feb 9, 2012
Last (family) name:
First (given) name:
I.D.#:
userid:
Question Marks
1
12
2
10
3
17
4
Fall 2014
STAT 334
Zhu, M
University of Waterloo
Statistics 334
Problem Set 1
Due: Thursday, October 2, 2014
1. Recall that, if X has density function f (x), then E(g(X) =
holds for discrete X.
i
h
1
Fall 2014
STAT 334
Zhu, M
University of Waterloo
Statistics 334
Problem Set 3
Due: Thursday, November 20, 2014
1. Suppose that independent trials, each of which is equally likely to have any of m poss
A'ssiammt I Salk-LBKS
1 (a X m Poiamnfﬁl] :30 FILE] 2 gt?" for .1: = 0,1,1.
"’ Since X is discrete. Eﬁgﬁxjj = Z gﬁxﬁhj. Therefan
1 “'4 1 A14
E<X+1)_ZJ-+1(EE)
1'=|:|
2%;{fiy
=§(e’~—1‘J, Billet"
Fall 2014
STAT 334
Zhu, M
University of Waterloo
Statistics 334
Problem Set 4
Due: Friday, December 5, 2014
1. Jock and Wayne are two brothers. Theyve just taken over as managers of two small KFC stor