7
Renewal Theory and Its Applications
1. D. R. Cox, Renewal Theory, Methuen, London, 1962.
2. W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II,
John Wiley, New York, 1966.
3. F. Hwang and D. Trietsch, A Simple Relation Betwee
STAT2303 & STAT3603 Probability Modelling
Class Test (28 Oct 2013)
1. There are two players who will take turn shooting independently at a target. Each shot by
player i hits the target with probability pi, i 1, 2. In each round, when a player hits the tar
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT3603&2303
2
Probability Modelling
Random Variables
Then random variables are real-valued functions defined on the sample space.
2.1
Discrete Random Variables
A discrete random
2015-16 Second Semester
MATH 2101 Linear Algebra I
Test 1
Name:
Group
For Markers Use Only
University No
Q1
Q2
Q3
Q4
Q5
Total
Important Notes:
Fill in your name (exactly as on student card), tutorial group number and university
number above.
Answer Ques
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT3603&2303 Probability Modelling
Assignment 1
Due : October 27, 2016
1. The joint probability mass function of X and Y , p (x, y), is given by
p (1, 1) = 19
p (1, 2) = 19
p (1,
2015-16 Second Semester
MATH 2101 Linear Algebra I
Test 2
Name:
Group
For Markers Use Only
University No
Q1
Q2
Q3
Q4
Q5
Total
Important Notes:
Fill in your name (exactly as on student card), tutorial group number and university
number above.
Answer Ques
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT3603&2303
4
4.1
Probability Modelling
Markov Chains
Introduction
A discrete state space stochastic process cfw_Xn : n = 0, 1, 2, . . . is called a Markov Chain (MC) if
P (Xn+1
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT3603&2303
5
Probability Modelling
The Poisson Process
5.1
The Exponential Distribution
A continuous random variable X is said to have an exponential distribution with parameter
Midterm solution
1. (a) Let T be the number of days to freedom.
E(T ) =
3
X
E(T |door i picked first) P (door i picked first)
i=1
=0.5[3 + E(T )] + 0.4[5 + E(T )] + 0.1(0)
=3.5 + 0.9E(T )
So, E(T ) = 3.5/0.1 = 35 days
(b)
2
E(T ) =
3
X
E(T 2 |door i picke
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT3603&2303 Probability Modelling
Assignment 1 Solution
1.
E [X | Y = i] =
3
X
xP (X = x | Y = i) =
x=1
3
X
xP (X = x, Y = i)
P (Y = i)
x=1
1
1
1
1 1 1
E [X | Y = 1] =
1 +2 +3
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT3603&2303 Probability Modelling
Assignment 2 Solution
1. (a) The cdf of Y = maxcfw_X1 , X2 is given by
FY (y) = P (Y y) = P (maxcfw_X1 , X2 y)
= P (X1 y, X2 y)
= P (X1 y)P (X
Tutorial 8
Pt
1. Let cfw_X(t), t = 0, 1, 2, . . . be a stochastic process defined as X(0) = 0 and X(t) =
i=1 Yi
for t 1, where Y1 , Y2 , . . . is a sequence of independently and identically distributed random
variables with distribution N (, 2 ).
(a) Sho
Tutorial 3
1. A Markov chain X0 , X1 , X2 , . . . has the transition probability matrix as follows:
1
2
3
1 1/4 3/4 0
2 1/3 1/3 1/3
3
0 1/4 3/4
(a) Determine the probabilities P (X1 = 2, X2 = 2|X0 = 1) and P (Xn+2 = 2|Xn = 1) where n
is a positive intege
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT3603&2303
3
Probability Modelling
Conditional Probability and Conditional Expectation
In Chapter 1 and Chapter 2, we have given some ideas and basic concepts of conditional pro
THE UNIVESRTIY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT3603&2303 Probability Modelling (2016-2017, 1st semester)
Instructor :
Class Times :
Instructors office Hours :
Teaching Assistant :
Tutorial Time :
Dr. J.K. Woo, RRS226, jkwoo
201516 Second Semester
MATH 2101 Linear Algebra I
Test 1 Report
A.
Statistics
Score Distribution
Score Range
2029
3039
4049
5059
6069
7079
8089
9099
100
No of students
2
2
6
7
7
9
13
12
2
Item statistics
Question
1
2
3
4
5
Mean score
15.20
14.33
14.20
16.
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT3603&2303
1
Probability Modelling
Introduction to Probability Theory
1.1
Sample Space and Events
Basic ideas of Probability Theory
Sample space S the set of all possible outco
5. Conditioning
5.1 Conditional expectation: A double expectation formula Take two
random variables X and Y. How to compute E[h(X,Y)]? Use
E[h(X,Y)] = Ecfw_E[h(X,Y)|X] (= Ecfw_E[h(X,Y)|Y].)
To understand the double expectation Ecfw_E[h(X,Y)|X], suppose gi
3. General random variables
A general rv for an experiment has a range that is not countable. A prime
example is:
Example Pick a point at random from the unit interval (0,1) and let X = number
picked. X can take any number between 0 and 1, and X is NOT a
6. Poisson processes
6.1 The Exponential random variables X is an an exponential () random
variable if Pcfw_X t| =1-expcfw_-t, t>0.
The density of X is f(t) expcfw_-tIcfw_t>0, so X is actually gamma (1; 1/).
The expected value, the variance, and the MGF o
ISOM 3540
Homework 2
Solution
Q1
(a) Among the Haves, expect
4
6
fraction of Yes answers
P( Yes comes from Qs ) + P( Yes comes from QsC )
=
=
4
6
2
1+ 0
6
4
6
(b) Among the Have-Not, expect
2
6
fraction of Yes answers
P( Yes comes from Qs ) + P( Yes comes
ISOM 3540 Homework 4 Solution
Question 1
Let T~Unif[0,1] be the train arrival time and N(t) be the number of passengers
arriving before time t. Then the number of passengers who catch the train is N(T). By
the independent assumption N(T)|T ~ Poisson(100T)
2. Discrete random variables
Experiment: Produces observations called outcomes
Sample space: S = set of all outcomes
Event: a collection of outcomes; events are subsets of S
Probability of an event is the chance that the event occurs
A random variable is
ISOM 3540
HW1 Solution
Q1) Problem 7 (p.16 in Ross)
(a) The number of ways that 3 boys and 3 girls can sit in a row:
= 6! = 720
(b) The number of ways that 3 boys and 3 girls sit in a row if the boys and the girls
are each to sit together:
BBBGGG
GGGBBB
=
Ems.
35500
Ill
The number of times that an individual contracts a cold in a given year is a Poisson
random variable with parameter A = 3. Suppose a new wonder drug (based on
large quantities of vitamin C) has just been marketed that reduces the Poisson
pa
Introduction to Probability
Experiment: Produces observations called outcomes
Sample space: S = set of all outcomes
Event: a collection of outcomes; events are subsets of S
Probability of an event is the chance that the event occurs
Example Flip a coin on
4. Moment generating functions and Limit theorems
4.1 Moment generating function
MX(t)=E[expcfw_tX], t in some interval
is called the MGF of X
Example
(i) X is Gamma (, 1/).
MX(t) = E[expcfw_tX] = [/()] x-1expcfw_-(-t)xdx
0
= 1/(1-t/), t<
(ii) X is Poisso