THE CHINESE UNIVERSITY OF HONG KONG
Department of Statistics
STAT3007: Introduction to Stochastic Processes
Exercises for Week 1: Solutions
1. (Chevalier de Meres Problem) The probability of having at least one six
in 4 throws of die is 1 (5/6)4 = 0.518 w
THE CHINESE UNIVERSITY OF HONG KONG
Department of Statistics
STAT3007: Introduction to Stochastic Processes
Problem Sheet 1
The deadline for this Problem Sheet is 5.30pm on Thursday 6th October.
Please hand in your answers to Dong Fangyuan in LSB 126 or L
THE CHINESE UNIVERSITY OF HONG KONG
Department of Statistics
STAT3007: Introduction to Stochastic Processes
Problem Sheet 1
The deadline for this Problem Sheet is 5.30pm on Thursday 8th October.
Please hand in your answers to Dong Fangyuan in LSB 126 or K
THE CHINESE UNIVERSITY OF HONG KONG
Department of Statistics
STAT3007: Introduction to Stochastic Processes
Problem Sheet 1
The deadline for this Problem Sheet is 5.30pm on Friday 3rd October. Please
hand in your answers to Dong Fangyuan in LSB 126 or Kwo
THE CHINESE UNIVERSITY OF HONG KONG
Department of Statistics
STAT3007: Introduction to Stochastic Processes
Problem Sheet 2
The deadline for this Problem Sheet is 5.30pm on Thursday 16th October.
Please hand in your answers Dong Fangyuan in LSB 126 or Kwo
THE CHINESE UNIVERSITY OF HONG KONG
Department of Statistics
STAT3007: Introduction to Stochastic Processes
Exercises for Week 9: Solutions
1. (Exercise 5.1.4 in Pinsky and Karlin)
(a) (t)k et /k!, for k = 0, 1, . . . .
(b) P (X(t) = n + kX(s) = n) =
s.
STA3007 Introduction to Stochastic Process!
(2014~2015 Semester 1)
Tutorial 3
(20141008 W9 @ LSB LT5, 20141009 H3 @ LSK LT2)
1.
More about Expectation
For any random variable X 0
o Discrete: E ( X ) P( X x) 1 FX ( x)
x 1
x 1
o Continuous: E ( X ) 1 F
THE CHINESE UNIVERSITY OF HONG KONG
Department of Statistics
STAT3007: Introduction to Stochastic Processes
Problem Sheet 3
The deadline for this Problem Sheet is 5.30pm on Friday 20th November.
Please hand in your answers to Dong Fangyuan in LSB 126 or K
STAT3007
Introduction to Stochastic Processes (1112)
Solution to Assignment 5
Chapter 5
Problem 3.1
If W1 > w1 , W2 > w2 , then there is no event occurring in the interval [0, w1 ], and there
may 0 or 1 event occurring in the interval (w1 , w2 ], which i
THE CHINESE UNIVERSITY OF HONG KONG
Department of Statistics
STAT3007: Introduction to Stochastic Processes
Exercises for After Week 4
1. (Problem 3.2.3 in Pinsky and Karlin) Let Xn denote the quality of the
nth item produced by a production system with X
THE CHINESE UNIVERSITY OF HONG KONG
Department of Statistics
STAT3007: Introduction to Stochastic Processes
Exercises for After Week 9
1. (Exercise 4.4.2 in Pinsky and Karlin) Consider
tion probability matrix is given by
0
1
0
0.1 0.4 0.2
P=
0.2 0.2 0.5
STAT3007
Introduction to Stochastic Processes (1112)
Solution to Assignment 3
Chapter 3
Problem 4.1
Method I
Let 1, 2, 3, 4, 5, 6, 7, 8 denote the dierent patterns of three tosses, HHH, HHT, HTH,
HTT, THH, THT, TTH, TTT, respectively. Then, we can get th
STA3007 Introduction to Stochastic Process!
(20152016 Semester 1)
Tutorial 1 Solution
1
Example 1
1.
Z Z
1=
Z
5
1
Z
f (x, y)dxdy =
y
1
x
0
x
( + cy)dxdy =
5
Z
5
(
1
1
2
+ cy)dy = + 12c
10
5
c = 1/20.
2.
Z Z
P(X + Y < 3) =
f (x, y)dxdy
Z
x+y<3
1 Z 3x
x
(
STAT3007 Introduction to Stochastic Process
(20152016 Semester 1)
Tutorial 7 Solution
(20151111 W6 @ LSB LT4, 20151111 W10 @ ERB 407)
Example 1
Let W11 be the waiting time to get X1 = 1 s.t. W11 Exp 11
Let W12 be the waiting time to get X2 = 1 s.
STAT3007 Introduction to Stochastic Process
(2015~2016 Semester 1)
Tutorial 9
(20151125 W6 @ LSB LT4, 20151125 W10 @ ERB 407)
1.
Fundamental Mathematics 1storder ODEs
Linear ODEs
dy
f ( x) y g ( x)  (*)
dx
Multiply the integrating factor to both si
STAT3007 Introduction to Stochastic Process
(20142015 Semester 1)
Tutorial 7
(20141111 W6 @ LSB LT4, 20141111 W10 @ ERB 407)
1
Poisson Related Distribution  Binomial Distribution
Thm: If X(t) P oi(t), then X(t1 )X(t2 ) = N Bin(N, tt12 ) for t1 t2
STAT3007 Introduction to Stochastic Process
(2015~2016 Semester 1)
Tutorial 10
(2015121 W6 @ LSB LT4, 2015121 W10 @ ERB 407)
1.
Pure Death Process
Def: A Markovian X (t ) is a pure death process with death rate
k if X (t ) follows the following postu
First Step Analysis. Extended Example
These notes provide two solutions to a problem stated below and discussed in lectures (Sections 1, 2). The difference between these solutions will be discussed in Section 3 where also
several examples of problems whic
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STAT3007 Introduction to Stochastic Pro ces.3Ps
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THE CHINESE UNIVERSITY OF HONG KONG
Department of Statistics
STAT3007: Introduction to Stochastic Processes
Problem Sheet 4
The deadline for this Problem Sheet is 5.30pm on Friday 9th December.
Please hand in your answers to Dong Fangyuan in LSB 126 or Li
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STAT3007 Introduction to Stochastic Proces
THE CHINESE UNIVERSITY OF HONG KONG
Department of Statistics
STAT3007: Introduction to Stochastic Processes
Exercises for Week 1
1. The 17th century French nobleman, Chevalier de Mere, was a keen gambler
who corresponded with some the times most brilliant
THE CHINESE UNIVERSITY OF HONG KONG
Department of Statistics
STAT3007: Introduction to Stochastic Processes
Week 2 Exercises
1. (Exercise 2.1.1 in Pinsky and Karlin) I roll a sixsided die and observe the
number N on the uppermost face. I then toss a fair
STA3007 Introduction to Stochastic Process
(2015~2016 Semester 1)
Tutorial 1
(20150923 W7 @ LSB LT4, 20150923 W10 @ ERB 407)
1. Review on Probability
Discrete
Continuous
Cumulative
Distribution Function
(c.d.f.)
FX ( x) P( X x)
Probability Density Fun
THE CHINESE UNIVERSITY OF HONG KONG
Department of Statistics
STAT3007: Introduction to Stochastic Processes
Exercises for Week 5
1. (Slide 8 of the Markov Chains  First Step Analysis notes) Consider the
Markov chain described by the transition probabilit
STAT3007: Introduction to
Stochastic Processes
First Step Analysis
Dr. John Wright
1
Simple First Step Analysis
A Markov Chain cfw_ has state space cfw_ , , ,
with transition matrix =
Let the time of absorption be
= min
=
We would like to find
= = =
STAT3007: Introduction to
Stochastic Processes
Markov Chains  Some Special Examples
Dr. John Wright
1
Markov Chains with I.I.D. R.V.s
Let denote a discrete valued random
variable (r.v.) whose values are nonnegative
integers and where = =
for
= , , and
STAT3007: Introduction to
Stochastic Processes
Introduction and Some Basics
Dr. John Wright
1
Introduction and Some Basics
The ord stochastic
ord for to guess
o es fro
the Greek
Stochastic means random or chance
A stochastic model is in contrast to a
d
STAT3007 Introduction to
Stochastic Processes
Markov Chains Long Run Behaviour
Dr. John Wright
1
As
We can prove (by induction) that for a probability
transition matrix =
where
< , < we have
=
Since
+
lim =
+
+
+
+
< , we have
+
+
2
As
Not all co
STAT3007: Introduction to
Stochastic Processes
Markov Chains Introduction
Dr John Wright
1
Markov Process
A Markov process is a stochastic process
with the property that, given its value at time
( ), the value at time ( ), for any > ,
is not influenced b