Tutorial 7: HW3 and Midterm
JIANG, Changkun
March 16, 2016
Outline
HW3-Q1: Independent Events
HW3-Q2: RVs and Expectation
HW3-Q3: PMF and Expectation
HW3-Q4: Bayes Formula
HW3-Q5: RVs and PMF
HW3-Q6: RVs and PMF
HW3-Q7: Bayes Formula
Tutorial 7: HW3 and M

ENGG2430B: Engineering Mathematics III
Spring 2016
Problem Set Assignment 3
Due: 6 March, 2016, 6pm
Prob. 1 (Independent Events) (10pts) Let the sample space be S = cfw_1, 2, 3, 4 with equally
likely outcomes. There are different definitions of events A a

ENGG2430B: Engineering Mathematics III
Spring 2016
Problem Set Assignment 3
Due: 6 March, 2016, 6pm
Prob. 1 (Independent Events) (10pts) Let the sample space be S = cfw_1, 2, 3, 4 with equally
likely outcomes. There are different definitions of events A a

Lecture 14
Sums of random variables
Sum of random variables
It is common that an experiment consists of many
trials, represented by random variables, # , , & .
We are interested in a new random variable defined
as the sum of these individual random vari

Lecture 25
Sample Mean and
the Limit Theorems
The sample mean
One important application of sums of random variables is
in estimating statistics of random variables.
Consider a random variable, say the cumulative GPA of a
random student in your college

Lecture 12
Expectation and Variance
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 weighted ave

Lecture 23
Covariance and correlation
Expectation of jointly distributed
random variables
Proposition: If and are jointly distributed
random variables with probability mass function
(, ), then
,
= - - , (, )
0
/
If and are jointly distributed random

Lecture 17
Continuous Random
Variables
Continuous versus discrete
Discrete random variables have finite or countably
many but infinite possible values.
But there are random variables with uncountable
number of possible values, such as the distance to
tr

Lecture 16
The Poisson
Random Variable Continued
Poisson Paradigm
Consider events, with $ the probability that event
occurs, = 1, , . If all the $ are small and the trials
are independent or weakly dependent, then the
number of these events that occur

Lecture 15
The Poisson
Random Variable
What do these events have in common?
The number of misprints on a page of a book
The number of people in a community who survive
to age 100
The number of wrong telephone numbers that are
dialed in a day
The numbe

Tutorial for Homework 2
Tutor: LIU Fang
Date: Feb 13, 2017
Question 1
Calvin will toss a fair coin for three times. We assume that the coin
tosses are independent at each time.
(1) What is the probability of getting exactly two tails in
succession?
(2) Wh

Lecture 21
Conditional PMF
Conditional PMF
As we define conditional probability for events, we can
also define conditional probability for random
variables.
Given a random variable with an PMF # and an
event , > 0, the conditional PMF of conditioned
on

Lecture 2
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 pick

Lecture 20
Exponential Random
Variables
Recall the Geometric Random
Variable
When we introduced the geometric random variable,
, we explained that it depends on only one
parameter .
Let us introduce another parameter, , to define a
new family of geometr

Lecture 6
The
inclusion-exclusion principle
and its application
Wing Shing Wong
Inclusion-exclusion principle
For two events, and
= + ()
How about three events, , and ?
There is an extended version:
Proposition 4:
= + +
+ ()
How to prove it
App

Lecture 24
Central Limit Theorem
Recall the Poisson Paradigm
Consider events, with 1 the probability that event
occurs, = 1, , . If all the 1 are small and the trials are
independent or weakly dependent, then the number of
these events that occur app

Lecture 15
The Poisson
Random Variable
What do these events have in common?
The number of misprints on a page of a book
The number of people in a community who survive
to age 100
The number of wrong telephone numbers that are
dialed in a day
The numbe

Tutorial 3
Introduction to Homework
Solutions
By Yang Lin
(Feb. 2017, @CUHK)
Email: yanglin.cuhk@gmail.com
Institute: Dept. of Information Engineering, CUHK.
Question 1 (Counting)
1.1 Question:
Say there are five persons A, B, C, D, E standing in a line,

Lecture 2
Learning how to count,
again
W S Wong
Why do we need to count in
probability?
I pick a Lindor from the brown box without looking.
Suppose
What is the chance that I pick the blue color one, which is my
favorite?
Intuitively, the chance of pick

Lecture 13
Expectation and Variance II
and
Joint PMFs
A useful Identity
For constants , ,
+ = )
Proof:
+ = +
= ) )
= ) ()
)
Standard deviation of is the square root of the
variance and denoted by:
= ()
Example:
Consider the previous coin-flipping

Lecture 18
Normal Random Variable
Normal or Gaussian Random
Variables
Arguably the most important continuous random
variable used in engineering
A random variable is normally distributed with
parameters and % if its density function is given
by:
1
1
1

Lecture 22
Jointly continuous random
variables
Joint distribution
All concepts of joint discrete random variables can
be carried over to continuous random variables with
suitable adaptation.
Let and two continuous random variables
defined on the same s

Tutorial 5
Discrete Random Variable
By Yang Lin
(Feb. 2017, @CUHK)
Email: yanglin.cuhk@gmail.com
Institute: Dept. of Information Engineering, CUHK.
Random Variable and Probability Mass Function
2 In reality, random variable is a real-valued function with

ENGG2430E Assignment 4
Due Date: 5pm, April 01, 2014
Q1.[15 points]: Consider 2m persons forming m couples who live together at a given time. Suppose that at some later time, the probability of each
person being alive is p, independent of other persons. A

ENGG2430C: Probability and Statistics for Engineers
(Spring 2017)
Problem Set Assignment 1
Due: January 26, 2017, 5pm
Prob. 1 (15pts) Say there are five persons A, B, C, D, E standing in a line, count how many
different linear arrangements for:
(a) (5pts)

ENGG2430C: Probability and Statistics for Engineers
(Spring 2017)
Problem Set Assignment 2
Due: February 11, 2017, 5pm
1. (20 points) Calvin will toss a fair coin for three times. We assume that the coin tosses
are independent at each time.
(1) What is th