The Hong Kong University of Science and Technology
probability
ELEC 2600

Fall 2013
The Hong Kong University of Science and Technology
Department of Electronic and Computer Engineering
ELEC2600 Fall 2012 Homework4
Due on Dec. 7
To obtain full marks, your answers should be well presented with necessary steps and
comments. This homework i
The Hong Kong University of Science and Technology
probability
ELEC 2600

Winter 2014
Elec210: Lecture 24
Discrete Time Random Processes
Sum Processes
ISI Processes
Elec210 Lecture 24
1
Example 9.14: Random Step Process
Let In be a Bernoulli random process with parameter p. Let
1
Dn = 2 I n 1 =
1
if I n = 1
if I n = 0
mD (n) = E[ Dn
The Hong Kong University of Science and Technology
probability
ELEC 2600

Winter 2014
Elec210: Lecture 13
Multiple Random Variables (RVs)
Joint probability mass function of two discrete RVs
Marginal probability mass function
Elec210 Lecture 13
1
Vector Random Variables
A vector random variable
X is a function that assigns a vector
of r
The Hong Kong University of Science and Technology
probability
ELEC 2600

Winter 2014
Elec 210: Lecture 7
Expectation of a random variable
Expected value of a function of a random variable
Variance of a random variable
Elec210 Lecture 7
1
Expected Value
Complete knowledge about the random variable is contained in
the probability mass f
The Hong Kong University of Science and Technology
probability
ELEC 2600

Winter 2014
Elec210: Lecture 20
Sums of random variables
Mean and variance of sample means
Laws of large numbers
Elec210 Lecture 20
1
Sums of Random Variables
For any set of random variables,
E
VAR
j
X j =
Xj=
j =1
n
E[ X
X 1 , X 2 ,., X n
j]
j
n
VAR( X
j)+
j
The Hong Kong University of Science and Technology
probability
ELEC 2600

Winter 2014
Elec210: Lecture 21
Central Limit Theorem
The PDF of sums of random variables
The characteristic function
Proof of the Central Limit Theorem
Elec210 Lecture 21
1
Central Limit Theorem
Suppose Xi for icfw_1,2,n are independent and identically distribu
The Hong Kong University of Science and Technology
probability
ELEC 2600

Winter 2014
Elec210: Lecture 14
Pairs of continuous random variables
Review of 2D functions, differentiation and integration
Joint cumulative distribution function
Joint density function
Elec210 Lecture 14
1
Two Random Variables
One random variable can be consi
The Hong Kong University of Science and Technology
probability
ELEC 2600

Winter 2014
Elec 210: Lecture 6
Random Variables
Equivalent Events
Di
Discrete
t Random
R d
Variables
V i bl
Probability mass function (pmf)
Elec210 Lecture 6
1
Random Variables
A random variable X is a function that assigns a number to every outcome of
an exper
The Hong Kong University of Science and Technology
probability
ELEC 2600

Winter 2014
Elec210 Lecture 11
Expectation
Variance
Important Continuous Random Variables
Elec210 Lecture 11
1
Review: Expectation
Interpretation
The average value of a random variable if we repeat the underlying
experiment a large number of times.
times
The li
The Hong Kong University of Science and Technology
probability
ELEC 2600

Winter 2014
Elec210: Lecture 25
Continuous Time I.S.I. Random Processes
Poisson Random Process
Random Telegraph Process
Shot Noise Process
Weiner Process
Elec210 Lecture 25
1
The Poisson Process
Consider the following sequence of random processes conditioned on
The Hong Kong University of Science and Technology
probability
ELEC 2600

Winter 2014
Elec 210: Lecture 2
Specifying Random Experiments
Sample spaces and events
Set
S t Operations
O
ti
The Three Axioms of Probability
Corollaries
Probability laws for picking events at random
Elec210 Lecture 2
1
Specifying Random Experiments
A random
The Hong Kong University of Science and Technology
probability
ELEC 2600

Winter 2014
Elec 210: Lecture 9
Important discrete random variables
Bernoulli
Binomial
Geometric
Poisson
Discrete uniform
MATLAB commands for plotting probability mass functions
and generating discrete random variables
Elec210 Lecture 9
1
The Bernoulli Random
The Hong Kong University of Science and Technology
probability
ELEC 2600

Winter 2014
Elec 210: Lecture 8
Conditional Probability Mass Function
Conditional Expected Value
Elec210 Lecture 8
1
Conditional Probability Mass Function
The effect of partial information about the outcome of a random
experiment on the probability of a discrete r
The Hong Kong University of Science and Technology
probability
ELEC 2600

Winter 2014
Elec 210: Lecture 4
Conditional Probability
Properties
Total Probability Theorem
Bayes Rule
Independence
Elec210 Lecture 4
1
Conditional Probability
Knowledge that an event B has occurred may alter the
probability that an event A has also occurred.
The Hong Kong University of Science and Technology
probability
ELEC 2600

Winter 2014
Elec210: Lecture 17
One function of two random variables
Discrete random variables
Continuous random variables
Using conditioning
Elec210 Lecture 17
1
One function of two discrete RVs
Suppose X takes integer values i.
Suppose Y takes integer values j.
The Hong Kong University of Science and Technology
probability
ELEC 2600

Winter 2014
Elec210: Lecture 19
Single Gaussian random variable
Pairs of jointly Gaussian random variables
Gaussian random vectors
Elec210 Lecture 19
1
Gaussian Random Variable
The Gaussian random variable is
used to model variables that tend
to occur around a ce
The Hong Kong University of Science and Technology
probability
ELEC 2600

Winter 2014
Elec210: Lecture 18
Random vectors
Joint distribution/density/mass functions
Marginal statistics
Conditional densities
Independence
Expectation
Elec210 Lecture 18
1
N Random Variables
An N dimensional random vector is
a mapping from a probability s
The Hong Kong University of Science and Technology
probability
ELEC 2600

Winter 2014
Elec 210: Lecture 10
Single random variables
Discrete, continuous and mixed type
Cumulative distribution function (cdf)
Probability density function (pdf)
Conditional cdfs and pdfs
Elec210 Lecture 10
1
Random Variables
A random variable X is a funct
The Hong Kong University of Science and Technology
probability
ELEC 2600

Winter 2014
Elec210: Lecture 16
Conditional Probability
Conditional Expectation
Elec210 Lecture 16
1
Conditional Probability Mass Functions
Suppose that X and Y are discrete RVs assuming integer values.
The conditional pmf of Y given X is
pY  X (k  j ) =
P [Y =
The Hong Kong University of Science and Technology
probability
ELEC 2600

Winter 2014
Elec210: Lecture 22
Definition of a Random Process
Specification of a Random Process
Elec210 Lecture 22
1
Definition of a Random Process
Definition: A random process or
stochastic process maps a
probability space S to a set of
functions, X(t,)
It assi
The Hong Kong University of Science and Technology
probability
ELEC 2600

Winter 2014
Elec210  Probability and Random Processes
in Engineering
Spring Semester 2009/10
Instructors: Bing ZENG
Office: Rm 2434
Tel: 23587058
GSM: 94181354
Email: [email protected]
Elec 210: Lecture 1
Course details
Motivations
Where is the diamond?
Intended l
The Hong Kong University of Science and Technology
probability
ELEC 2600

Winter 2014
Elec210: Lecture 15
Independence
Joint Moments
Elec210 Lecture 15
1
Independence
Definition: Two random variables X and Y are said to be
independent or statistically independent if for any events, AX
and AY, defined in terms of X and Y respectively,
re
The Hong Kong University of Science and Technology
probability
ELEC 2600

Winter 2014
Elec210: Lecture 23
Mean and Correlation/Covariance Functions
Multiple Random Processes
Elec210 Lecture 23
1
Mean and Variance Functions
Mean
mX (t ) = E[ X (t )] = xf X ( t ) ( x) dx
Variance
2
Var[ X (t )] = E ( X (t ) mX (t ) ) = ( x mX (t ) 2 f X (
The Hong Kong University of Science and Technology
probability
ELEC 2600

Winter 2014
Elec 210: Lecture 3
Computing Probabilities using Counting Methods
Applicable for finite sample space
Typical examples: selecting balls
balls from an urn
urn
More general interpretation: sampling from a population
o With or without replacement
o With
The Hong Kong University of Science and Technology
probability
ELEC 2600

Winter 2014
ELEC 2600: Probability and Random
Processes in Engineering
Part I: Basic Probability Theory
Lecture 1: Course Introduction
Lecture 2: Build a Probability Model
Lecture 3: Counting Method
Lecture 4: Conditional Probability & Independence
Lecture 5: Sequent
The Hong Kong University of Science and Technology
probability
ELEC 2600

Winter 2014
ELEC 2600: Probability and Random
Processes in Engineering
Part III: Multiple Random Variables
Lecture 13: Pairs of Discrete Random Variable
Lecture 14: Pairs of Continuous Random Variable
Lecture 15: Independence, Joint Moments
Lecture 16: Conditional Pr
The Hong Kong University of Science and Technology
probability
ELEC 2600

Fall 2012
The Hong Kong University of Science and Technology
Department of Electronic and Computer Engineering
ELEC 2600 Fall 2012 Homework2
Due at 18:00 on Oct. 26 (2012)
To obtain full marks, your answers should be well presented with necessary steps and comment
The Hong Kong University of Science and Technology
probability
ELEC 2600

Fall 2012
The Hong Kong University of Science and Technology
Department of Electronic and Computer Engineering
ELEC 2600 Fall 2012 Homework2
Due at 18:00 on Oct. 26 (2012)
To obtain full marks, your answers should be well presented with necessary steps and comment
The Hong Kong University of Science and Technology
probability
ELEC 2600

Fall 2012
The Hong Kong University of Science and Technology
Department of Electronic and Computer Engineering
ELEC2600 Fall 2012 Homework3
Due on Nov. 26
To obtain full marks, your answers should be well presented with necessary steps and
comments. This homework
The Hong Kong University of Science and Technology
Probability and Random Processes in Engineering
ELEC 2600

Fall 2017
The Hong Kong University of Science and Technology
Department of Electronic and Computer Engineering
ELEC2400
ELECTRONIC CIRCUITS
SPRING 2014/15
LAB 2 PSpice Circuit Simulation
1. Objective
To study PSpice simulation of circuits and verify the network the