Lecture 10
Random Variables
Wing Shing Wong
Real-valued function of a variable
Given a set, , a real-valued function of assigns a
real value to each and every element of .
For example, = 2 + 3, = 2 , , are
functions defined on the set of real numbers.

ENGG2430C: Engineering Mathematics III
Spring 2014
Problem Set Assignment 6
Due: April 16th, 2014, 6pm
Problem 12, 14, and 15 in http:/athenasc.com/CH7-prob-supp.pdf. (Each problem is
worth 20 points.)
Prob. 4. Let X be a Gaussian random variable with mea

ENGG2430C Lecture 13: Central Limit Theorem
and Hypothesis Testing
Minghua Chen (minghua@ie.cuhk.edu.hk)
Information Engineering
The Chinese University of Hong Kong
Reading: Ch. 5.4 and Ch. 8 of the textbook.
M. Chen (IE@CUHK)
ENGG2430C lecture 13
1 / 22

Hypothesis Testing and Maximum Likelihood
Estimation
Minghua Chen (minghua@ie.cuhk.edu.hk)
Information Engineering
The Chinese University of Hong Kong
Reading: Ch. 8 and 9.3 of the textbook.
M. Chen (IE@CUHK)
ENGG2430C lecture 12
1 / 21
Review: Hypothesis

Prob. 1 [10 pts] (Lecture Note and Homework Revisit)
(a) (5 pts) Let A and B be two events with positive probabilities. Prove that if they
are disjoint, then they cannot be independent. (Explanation using Venn diagram
is ne.)
(b) (5pts) Let A and B be t

ERG2430C: Engineering Mathematics III
midterm solution
Solution to Problem 1 (10pts)
(a) (5pts)
1) On the one hand, we have P (A) > 0 and P (B ) > 0, which implies that P (A)P (B ) > 0.
2) On the other hand, since A and B are disjoint, we have P (A B ) =

ENGG2430c: Engineering Mathematics III
Spring 2014
Solution for homework 4
Date: Mar 24th, 2014
Prob. 1 Solution By Bayes Theorem and total probability theorem,
P (A|x
1
P (x 4 |A)P (A)
1
)=
4
P (x 1 )
4
1
P (x 4 |A)P (A)
1
P (x 4 |A)P (A) + P (x 1 |B )P

ENGG2430C: Engineering Mathematics III
Spring 2014
Problem Set Assignment 4
Due: March 19th, 2014, 6pm
Problem 13 in http:/athenasc.com/CH3-prob-supp.pdf.
Prob. 2. (40pts)
(a) (30pts) Problem 9 in http:/www.athenasc.com/CH3-prob-supp.pdf. (Note:
15 pts fo

ENGG2430C: Engineering Mathematics III
Spring 2014
Solution for homework 3
Date: Feb 26nd, 2014
Prob. 1. Solution:
By denition of independence, we need to show
P (A (B C ) = P (A)P (B C ).
()
To prove this, we will use the following equality which we have

ENGG2430C: Engineering Mathematics III
Spring 2014
Problem Set Assignment 5
Due: April 2nd, 2014, 6pm
Problem 20 and 21 in http:/athenasc.com/CH3-prob-supp.pdf. (Each problem is worth
30 points.)
Problem 3, 5, and 10 in http:/athenasc.com/CH7-prob-supp.pd

ENGG2430c: Engineering Mathematics III
Spring 2014
Solution for homework Five
Date: April 2nd, 2014
Solution to Problem 20 in CH3 (30)
(a) Dene the following events:
A = cfw_One day is sunny,
B = cfw_One day is rainy.
Denote r.v. X as your driving time in

Lecture 1b
Learning how to count,
again
W S Wong
Why do we need to count in
probability?
Suppose I pick a Lindor from the brown box without looking. What is the
chance that I pick the blue color one, which is my favorite?
Intuitively, the chance of pic

Lecture 3
How to define probability?
Probability is an estimate of the
likelihood of a random event.
But
How to estimate something that
cannot be predicted?
Wing Shing Wong
Recall the Lindor example
How should we define the probability of picking the bl

Lecture 12
Expectation and Variance II
and
Joint PMFs
Wing Shing Wong
A useful Identity
For constants , ,
+ = 2
Proof:
+ = +
= 2 2
= 2 ()
2
Standard deviation of is the square root of the
variance and denoted by:
=
()
Example:
Consider the previ

Lecture 11
Expectation and Variance
Wing Shing Wong
Expectation
If is a discrete random variable with probability mass
density () the expectation or expected value, or the
mean value or average value of , is defined by:
=
()
: >0
That is, it is a wei

Lecture 8
Bayes Formula and
Independent Events
Wing Shing Wong
Revisiting the Monty Hall
Problem
Game Rule
From S. Lucas, J. Rosenhouse, A. Schepler, The Monty Hall, Reconsidered
(Classic Monty). You are a player on a game show and are
shown three ident

Lecture 9
Independent Events
Wing Shing Wong
Network connectivity example (p.41 textbook)
Consider a computer network connecting node A to
node B as a graph
The weight on an edge represents the probability that
the link is up, in other words, the conne

Lecture 7
Bayes Formulas
Wing Shing Wong
Recall the conditional probability
formula
Given two events, and , we have defined the
probability of given as
()
=
()
But we could also consider the conditional probability of
given has occurred.
()
=
()
We

Lecture 6
Conditional Probability
Wing Shing Wong
Your true friend or not
A friend told you that he learned a novel way to play a
game involving 3 chips, 1 is blue on both sides, 1 is
red on both sides, the third, blue on one, red on the
other.
The chi

Lecture 4
Properties of the Probability
Function
Wing Shing Wong
Simple Propositions
Proposition 1: = 1
E.g. If = , , , then
,
=
= 1 ( )
Proof of Proposition:
1 = = = + ( ).
Wing Shing Wong
Simple Propositions
Proposition 2: If , then .
E.g. If =

Lecture 5
The
inclusion-exclusion principle
and its application
Wing Shing Wong
Extension of Axiom 3
Axiom 3: For any sequence of mutually exclusive events
1 , 2 ,
=
=
For any two events, and
= + ()
For three events, , and
= + +
+ ()
Wing Sh

Lecture 2
Combinations and
the Binomial Theorem
Example from poker ordering
The highest hand is a straight flush which consists of cards of
the same suit in sequence:
E.g. 5 4 3 2 A, 6 5 4 3 2, , K Q J 10 9 ,
A K Q J 10
Hence there are 10
1
4
= 40 such

ENGG2430c: Engineering Mathematics III
Fall 2013
Solution for homework One
Date: Jan 22nd, 2014
Prob. 1. Solution. Lets rstly denote D = A B , we can get P (D) = P (A) + P (B )
P (A B ) by the formula. Then
P (A B C ) = P (D C )
= P (D) + P (C ) P (D C )

ENGG2430c: Engineering Mathematics III
Spring 2014
Solution for homework Two
Date: Feb 18nd, 2014
Solution to Problem 18 (10)
First we dene the following events,
1s
0s
1d
0d
= cfw_The bit is 1 when storing,
= cfw_The bit is 0 when storing,
= cfw_The bit i

ENGG2430C: Engineering Mathematics III
Spring 2014
Problem Set Assignment 2
Due: February 18th, 2014, 6pm
Problems 18, 23, and 29 in http:/athenasc.com/CH1-prob-supp.pdf. (Each problem is
worth 10 points.)
Prob. 4 (10pts) Envelop Paradox. There are two en

ENGG2430C Lecture 8: Continuous RV, PDF,
and CDF
Minghua Chen (minghua@ie.cuhk.edu.hk)
Information Engineering
The Chinese University of Hong Kong
Reading: Ch. 3 of the textbook
M. Chen (IE@CUHK)
ENGG2430C lecture 8
1 / 26
Review: Binomial R.V. and Poisso

ENGG2430C Lecture 9: Derived Distribution and
Generating Random Variable
Minghua Chen (minghua@ie.cuhk.edu.hk)
Information Engineering
The Chinese University of Hong Kong
Reading: Ch. 3 of the textbook
M. Chen (IE@CUHK)
ENGG2430C lecture 9
1 / 20
Review:

ENGG2430C Lecture 10: Inequalities, Law of
Large Number, and Central Limit Theorem
Minghua Chen (minghua@ie.cuhk.edu.hk)
Information Engineering
The Chinese University of Hong Kong
Reading: Ch. 5 of the textbook.
M. Chen (IE@CUHK)
ENGG2430C lecture 10
1 /

Example RV, Total Expectation Theorem, and
Application Examples
Minghua Chen (minghua@ie.cuhk.edu.hk)
Information Engineering
The Chinese University of Hong Kong
Reading: Ch. 2.5-2.7 of the textbook
Notice: Midterm next Thursday evening 7-9pm in LSK_LT5.

ENGG2430C Lecture 3:
Conditional Probability and Independence
Minghua Chen (minghua@ie.cuhk.edu.hk)
Information Engineering
The Chinese University of Hong Kong
Reading: Ch. 1.3 and 1.5 of the textbook
M. Chen (IE@CUHK)
ENGG2430C lecture 3
1 / 21
Review: P