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) = ex
and fY (y) = ey
for x, y > 0.
The goal here is t
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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 average to reach (or
hit) either A or C?
&
Stat 334 |
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
differentiation we can solve for the p(d)f.
2. If U is a linear c
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
an interval is a normal random variable.
Example: Let
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 of the
self-similarity property:
Rescale to smaller and
Lecture 18
16th July 2013
Interpreting U(t): Suppose c=2, then as t
goes from 0 to 1, U(t) looks like Z(t)
between 0 and multiplied by a factor of
2. In other words the plot of U(t) as t goes
from 0 to 1 looks exactly like a standard
Brownian motion.
Th
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Example S
There are n distinct prizes (e.g., $10, $100, $25, .). Sequentially,
you will be presented one prize at a time, whereupon you must
either accept or reject it. If you accept, thats the prize you will
walk away with; if you reject, you cannot
UNIVERSITY OF WATERLOO
STAT 334
Probability Models for Business and Accounting
Fall 2016
Mu Zhu, PhD
Professor, Department of Statistics & Actuarial Science
Description
This is a special course designed for a relatively small group of students. Roughly
sp
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Example 1
A deck contains k regular cards and a joker. You and I take
turns to draw a card from this deck until someone draws the joker.
You go first. Whats the probability that you will get the joker?
&
Stat 334 | Lecture 10
2008-16 by Mu Zhu, PhD
1
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 you identify the distribution?
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 expected number of different
types that are contained in a s
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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 behavior of this Markov chain?
Remark If the first row of the
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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 reasonable model?
(b) Why is the negative binomial distribution
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
=
=
e
[e 1]
1 e
and
E erT =
=
=
Z
ert
Z0
1 t
t
e dt
()
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) = P (X x) = 1 ex . Think of X as
the amount of time yo
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, indicating clearly that you have down so.
Question
1
Possib
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, indicating clearly that you have down so.
Question
1
Possib
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
Probability
A
0.1
B
0.4
C
0.3
D
0.2
Give the expression
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 least 2 events
in the second hour?
c)What is the probabi
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 known pdf of X to evaluate the
probability of this event
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Example 0
Suppose
X|N Binomial(N, p)
and N Poisson().
Whats the marginal distribution of X?
Remark For example, N may be the number of customers and
X may be the number of sales.
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Stat 334 | Lecture 11
2008-16 by Mu Zhu, PhD
1
%
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Example 1
Suppos
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YUM ~— 2 + Y so Emw-ﬂ : Z+£Cﬂ
“KILL ‘- 5? Y so Emlyn]: [yam]
X
[r[ - :5 -; +‘ 0s5 30 : 0‘6 é
9f+3 O~S
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: %(ZkECY1)+%<5*6£\7/1> * l5“)
ELYLJ‘PSK -_ 0.3(52
as EH1] : EEEHLIH]
: w + 4m + my w
STAT 334 - Spring 2015 - Assignment 1
Due Friday June 5 at start of tutorial
This assignment may be completed in pairs. Only one person should submit the paper. Both
names and ID numbers should be on it. Both will receive the same grade.
1. Let X be a neg
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
9
5
12
Total
60
1
Score
1
1. Suppose we toss a fair co
University of Waterloo
STAT 334 Test 1
Spring 2015
j
First (Given) Name: SQi—«UT i9 M5 Last (Family) Name:
Student ID #: userid:
Marking Scheme:
Instructions:
1. Please read questions carefully and show your work in the space provided.
2. This pap
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
.
(a) Suppose X Poisson(). Calculate E X+1
R
g(x)f (x)d
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 possible outcomes,
are performed until the same outcome occ
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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, i.e., X0 = 1.
(a) Find the distribution of Xn , for n =
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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 x y)!
(a) What is the marginal distribution of X?
(b) W
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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 Var(Y )? That is, how much variability is there for the
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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 Pijn = 0 for all n > 0, then
P [(ever go to state j)|(sta