Chapter 2: Power Spectral Density of
Stationary Processes
School of Science
BUPT
February 15, 2009
2.1
Power Spectral Density of Stationary Processes
Introduction:
How to calculate the power of an
electronic signal, which is regarded as
some stochastic pr
Chapter 4: Expectation
School of Science
BUPT
February 15, 2009
4.7
The Sample Mean
The average of a random sample of i.i.d.
random variables is called their sample mean.
The sample mean is useful for summarizing
the information in a random sample in much
Chapter 4: Expectation
School of Science
BUPT
February 15, 2009
4.6
Covariance and Correlation
Given
(X , Y ) f (x , y ),
How much the two random variables, X and
Y , depend on each other?
A useful measure is Covariance (or
Correlation), which captures a
Chapter 4: Expectation
School of Science
BUPT
February 15, 2009
4.5
The Mean and The Median
Both the mean of a distribution and the
median are measures of central location for a
distribution.
How do you distinguish them?
4.5
The Mean and The Median
Both t
Chapter 4: Expectation
School of Science
BUPT
February 15, 2009
4.4
Moments
Question:
E (X 5 ) =?,
What is the distribution of
X1 , . . . , Xn are i.i.d.
n
i =1 Xi ,
A sucient tool: moment generating
function(t ) = E (e tX )
where
4.4
Moments
Question:
E
Chapter 4: Expectation
School of Science
BUPT
February 15, 2009
4.3
Variance
Let X , Y satisfy
E (X ) = 2,
Pr(Y = 2) = 1.
Question:How to distinguish between them?
Two useful measures:
variance!
The standard deviation is the square-root
of the variance.
4
Chapter 4: Expectation
School of Science
BUPT
February 15, 2009
4.2
Properties of Expectations
In this section we present some results
that simplify the calculation of
expectations for some common
functions of random variables.
4.2
Properties of Expectati
Chapter 4: Expectation
School of Science
BUPT
February 15, 2009
4.1
The Expectation of a Random Variable
The distribution of a random variable X : all of
the probabilistic information about X .
The average value or expected value: where
we expect X to be
Chapter 3: Random Variables and
Distributions
School of Science
BUPT
February 15, 2009
3.9
Functions of Two or More Random Variables
Given the distribution of a random variable X
and Y , how do you know the distribution of its
function: r (X , Y ), or the
Chapter 3: Random Variables and
Distributions
School of Science
BUPT
February 15, 2009
3.8
Functions of a Random Variable
Given the distribution of a random variable X ,
how do you know the distribution of its
functiona new random variable.
X f
1
X
?
g (
Chapter 3: Random Variables and
Distributions
School of Science
BUPT
February 15, 2009
3.7
Multivariate Distributions
The results for two random variables X and Y
The results for an arbitrary nite number n of
random variables X1 , . . . , Xn .
In general,
Chapter 3: Random Variables and
Distributions
School of Science
BUPT
February 15, 2009
3.6
Conditional Distributions
Conditional distributions are the generalization
of the concept of conditional probability:
Pr (A|B ) =
Pr (AB )
Pr (b )
Contents
Discrete
Introduction
The Bernoulli and . . .
The Hypergeometric . . .
The Poisson Distribution
The Normal Distribution
The Central Limit . . .
The Bivariate Normal . . .
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Beijing University of Posts
Probability Theory and Stochastic Processes
Yongjiang Guo
[email protected]
Science school, BUPT
Random Variables and Discrete Distributions
We use probability to measure an event. But the
event is not that mathematical.
Translate the event by a spec
Probability Theory and Stochastic Processes
Yang Jiankui
[email protected]
Science school, BUPT
February 15, 2009
Introduction to the denition
Relationship between two events
Union, intersection . . .
Update your decision when certain events are observed.
Probability Theory and Stochastic Processes
School of Science BUPT
February 15, 2009
History
Origin
Dice in 5 thousand years ago.
Theory started by the French mathematicians Blaise Pascal
(16231662) and Pierre Fermat (16011665).
References.
History
Origin
Stochastic Processes and Their Statistical
Description
School of Science, BUPT
Basic Denitions
Contents:
Denitions
The Probability Distribution Function of
Stochastic Process
Mean; Autocorrelation; Autocovariance
Two-Dimensional Processes
The Moment of th