ELEC 2600:
Probability and Random Processes in
Engineering
Spring 2012
Instructor: Prof. Jun ZHANG
Office: Rm 2448 (Lift 25-26)
Tel: 2358 7050
Email: [email protected]
Webpage: http:/www.ece.ust.hk/~eejzhang/
1
Elec 2600: Lecture 1
Course Details
Models i
Elec2600: Lecture 16
Conditional Probability
Conditional Expectation
Elec2600 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 k
Elec2600: Lecture 17
One function of two random variables
Discrete random variables
Continuous random variables
Using conditioning
Thus far, for Z = g(X,Y), with X and Y
random variables, we know how to
compute the moments of Z.
But how do we compute the
Elec2600: Lecture 18
Random Vectors
Joint distribution/density/mass functions
Marginal statistics
Conditional densities
Independence and Expectation
http:/www.cs.princeton.edu/~cdecoro/eigenfaces/
Elec2600 Lecture 18
1
N Random Variables
An N dimens
Elec2600: Lecture 19
Single Gaussian Random Variable
Gaussian Random Vectors
Elec2600 Lecture 19
1
Gaussian Random Variable
The Gaussian random variable is
used to model variables that tend
to occur around a certain value, m,
called the mean.
This ran
Elec2600: Lecture 20
Sums of Random Variables
Mean and Variance of Sample Means
Useful Inequalities
Laws of Large Numbers
Elec2600 Lecture 20
1
Sums of Random Variables
For any set of random variables, X 1 , X 2 ,., X n
E
VAR
j
X j
Xj
j 1
n
E[ X
j
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
http:/www.mathsisfun.com/data/quincunx.html
1
Central Limit Theorem
Any distribution
Suppos
Elec2600: Lecture 22
Definition of a Random Process
Specification of a Random Process
Elec2600 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
Elec2600: Lecture 23
Mean and Variance
Correlation and Covariance Functions
Multiple Random Processes
Elec2600 Lecture 23
1
Mean and Variance Functions
Mean
mX (t ) E[ X (t )] xf X ( t ) ( x) dx
Variance
X (t ) mX (t ) 2 ( x mX (t ) 2 f X (t ) ( x)dx
Elec2600: Lecture 24
Discrete Time Random Processes
Sum Processes
ISI Processes
Elec2600 Lecture 24
1
Sum Random Processes
Definition: A sum process Sn is obtained by taking the sum of
all past values of an i.i.d. random process Xn, i.e.,
n
S n X i X
Elec210: Lecture 25
Continuous Time I.S.I. Random Processes
Poisson Random Process
Additional Random Processes (FYI)
Random Telegraph Process
Shot Noise Process
Weiner Process
Elec210 Lecture 25
1
The Poisson Process
Consider the following sequence of
Elec2600: Lecture 26
Stationary Random Processes
Wide Sense Stationary (WSS) Random Processes
Elec2600 Lecture 26
1
Stationary Random Processes
Definition: A process is stationary if the joint distribution of any set of samples
does not depend on the pl
Elec2600: Lecture 15
Independence
Expectation of Function of 2 Variables
Joint Moments
Elec2600 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, de
Elec2600: Lecture 14
Pairs of continuous random variables
Review of 2D functions, differentiation and integration
Joint cumulative distribution function
Joint density function
Elec2600 Lecture 14
1
Two Random Variables
One random variable can be consid
Elec2600: Lecture 13
Multiple random variables (RVs)
Joint probability mass function of two discrete RVs
Marginal probability mass function
Elec2600 Lecture 13
1
Vector Random Variables
A vector random variable X is a function that assigns a vector
of
ELEC 2600 Lecture 2: Build a Probability Model
Specifying Random Experiments
Sample spaces and events
Set Operations
The Three Axioms of Probability
Corollaries
Probability Laws for Assigning Probabilities
Elec2600 Lecture 2
1
Specifying Random Expe
Elec 210: Lecture 3
Computing Probabilities using Counting Methods
Sample Size Computation and Examples
Probabilities and Poker!
Elec2600 Lecture 3
1
Computing Probabilities Using Counting Methods
In experiments where the outcomes are equiprobable, we
Elec 2600: Lecture 4
Conditional Probability
Properties
Thus far, we have looked at the probability
of events occurring individually, without
regards to any other event.
However, if we KNOW that a particular
event occurs, then how does this change the
p
Elec 2600: Lecture 5
Sequential Experiments
Bernoulli trials
Binomial probability law
*Multinomial probability law
Geometric probability law
Sequences of dependent experiments
Example: Bean Machine Game!
Elec2600 Lecture 5
1
Sequential Experiments
Elec 2600: Lecture 6
Random Variables
Equivalent events
Discrete Random Variables
Probability mass function
Elec2600 Lecture 6
1
Random Variables
A random variable
X is a function that assigns a number to every
outcome of an experiment.
The function
Elec 2600: Lecture 7
Expectation of a random variable
Expected value of a function of a random
variable
Variance of a random variable
Moments of a random variable
The human brain weighs
1500 grams, on average.
Elec2600 Lecture 7
1
Interesting Fact Ein
Elec 210: Lecture 8
Conditional Probability Mass Function
Conditional Expected Value
Conditional Poker Hand
Elec210 Lecture 8
1
Conditional Probability Mass Function
The effect of partial information about the outcome of a random
experiment on the pro
Elec 2600: Lecture 9
Important discrete random variables
Summary of variables you know:
Bernoulli
Binomial
Geometric
Discrete Uniform
New random variable: Poisson
MATLAB commands for plotting probability mass functions
and generating discrete random v
Elec 2600: Lecture 10
Single random variables: discrete, continuous and mixed
Continuous R.V. and Cumulative Distribution Function (CDF)
Probability Density Function (PDF)
Conditional CDFs and PDFs
Elec2600 Lecture 10
1
Random Variables: Review
A random
Elec2600 Lecture 11
Expectation of Continuous Random Variables
Variance of Continuous Random Variables
Important Continuous Random Variables
Elec2600 Lecture 11
1
Review: Expectation
Interpretation
The average value of a random variable if we repeat
Elec2600 Lecture 12
MATLAB functions for continuous random variables
Functions (Transformations) of a Random Variable
Elec2600 Lecture 12
1
MATLAB Functions for Continuous RVs
Generating m by n arrays of random samples
unifrnd(a,b,m,n)
exprnd(1/lambda,
SIU Chung Pan
LANG4031 T04
The optimum charging mechanism for lithium-ion battery devices
With the extensive use of electronic devices such as cell
phones, laptops and digital cameras, lithium-ion batteries have
been commonly employed in these devices bec