Lecture 4
th
12 May 2016
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 e
Bond Ratings
Analysis Using
Markov Chains
By Eric Kamushana, Kwabena
Gyasensir & Thomas Joseph
Introduction
What is Bond Credit Rating?
A bond rating is a grade to represent the
credit quality of the bond
Financial firms face different kinds of risks
1.
Lecture 6
th
19 May 2016
What if out variables are Discrete in nature?
Example: Suppose Y~(, ), find E[Y].
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 t
Chapter 2: Multivariate Distributions
Specific Discrete Multivariate Random Variables
Multinomial Distribution:
Extension of a Bernoulli trial, where now we
have more than 2 possible outcomes
BUT the trials are still assumed to be
independent and,
The
https:/www.coursehero.com/file/13799883/hw2solnpdf/
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Fall 2016
STAT 334
Zhu, M
University of Waterloo
STAT 334
Assignment #3
( Distributed: Sunday, November 6, 2016; Due: Tuesday, November 15, 2016 )
1. Suppose that independent trials, each of which is equally likely to have any of m possible
outcomes, are
Fall 2016
STAT 334
Zhu, M
University of Waterloo
STAT 334
Assignment #3
( Distributed: Sunday, November 6, 2016; Due: Tuesday, November 15, 2016 )
1. Suppose that independent trials, each of which is equally likely to have any of m possible
outcomes, are
Lecture 10
nd
2 June 2016
Computing Probabilities by Conditioning:
We can also use this conditioning approach to
compute probabilities.
Example:
An automobile insurance company classifies each of
its policy holders as being of one of the types =
1, .
Lecture 9
th
30 May 2016
Hence the Var(X|Y=y) is
Example:
Suppose the joint density of X and Y is given by
6 2 ,
0 < < 1, 0 < < 1
, =
0
otherwise
Compute the conditional expectation of X given that = ,
where 0 < < 1
Lecture 8
th
26 May 2016
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
Lecture 7
th
24 May 2016
Recall:
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?
Approaches:
1.
Chapter IV Markov Chains
T can be discrete, cfw_0, 1, 2, or continuous,
(0, ).
And at any time t, the value of Xt specifies the
state of the process.
For instance, = means that at time t, the
process is in state i.
Where the state space can also be di
Chapter V: The Poisson Process
For example:
If we suppose that the price of an asset is
subject to a series of relatively large shocks or
jumps that occur randomly over time. We can
use the Poisson process to model when these
jumps occur and the size of
Chapter 3: Conditional probability and Random
Variables
Hence the Var(X|Y=y) is
Example:
Suppose the joint density of X and Y is given
by
6 2 ,
, =
0
0 < < 1, 0 < < 1
otherwise
Compute the conditional expectation of X
given that = , where 0 < < 1
C
STAT 334 - Probability Models for Business and Accounting
Spring 2016
Instructor:
Dina Dawoud M3 3126 ddawoud office hours: 1:30-3pm T, and 2:30-4pm Th in M3 3106
Textbook/Course Notes: The primary source will be the slides posted on Learn and the notes
g
Stat 334
Lecture 1
3rd May 2016
Dina Dawoud
M3 3126
Lecture Outline:
Introduce Myself
Introduce the Course: Course Syllabus
Course Outline
Chapter 1: Motivation and Review
Motivation and Course Overview
Multivariate distributions (discrete and con
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https:/www.coursehero.com/file/12604257/Tutorial-2-Solution
STAT 334 Spring 2016 Project Information
The project in this course is intended to stimulate creative thinking and to develop the skills of
independent modelling and analysis of real life problems. It includes:
Choosing a topic of interest to you, buildin
”total-93‘“
STAT 334 Spring 2016 Tutorial 2
Names:
IDs:
bQuestion 1; X' N QXP (9‘ 6W5)
X2 "’ EXP (6% 6w»)
9"!- Suppose the length of time a new, long-life bulb burns is exponentially distributed with a mean
9% ‘J of 5 years. A homeowner has two such bulbs
Names:
IDS:
' Question 1: _
Suppose that large jumps in the value of an asset occur according to a Poisson process with an
average of 1.2 per day. Let X denote the number of jumps during an interval of length t days, that
[X is X ~ Pois(1.2t).
a) What is
STAT 334
Spring 2016
Assignment 1
Due: Friday, 27th May 2016 on Learn by 12pm
Question 1 (9 marks):
Let the probability density of X be given by
() = cfw_
(4 2 2 ),
0,
0<2
otherwise
a) (3) What is the value c?
1
3
b) (3) cfw_2 < < 2 = ?
c) (3) solve for E
Lecture 2
th
5 May 2016
Recall:
Some Important Random Variables:
Discrete Random Variables
Binomial Random Variables:
We have a series of Bernoulli trials that have the
following properties:
1.
2.
3.
4.
Two Outcomes (Success(S) vs. Failure(F)
Independe
Stat 334
Lecture 1
1st May 2012
Dina Dawoud
M3 3126
Motivation and Course Overview
Multivariate distributions (discrete and
continuous random variables, moment
generating functions.)
Conditional Probability and Random Variables
Markov Chain:
Named aft
Lecture 3
th
10 May 2016
Recall:
The Exponential Random Variable
T~ exp()
Where does this come from?
Suppose we have a Poisson Process where jumps
occur at a rate per unit time.
Then consider a random variable T which represents
the time until the F
Chapter 6
Brownian Motion
Now we consider an important Stochastic
process that has continuous state space
(; ) and operates on continuous time,
t>0.
To motivate the Brownian Motion:
Consider a random walk as discussed in Chapter 4
(i.e. Markov chain)
Z