Elec210: Lecture 26
Stationary Random Processes
Wide Sense Stationary (WSS) Random Processes
Elec210 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 placem
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
Elec210: Lecture 24
Discrete Time Random Processes
Sum Processes
ISI Processes
Elec210 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 1
Elec210: Lecture 23
Mean and Variance
Correlation and 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
E X t m t
VarX t E X t m X t x m X t f X t x dx
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 assign
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
Elec210: Lecture 20
Sums of Random Variables
Mean and Variance of Sample Means
Useful Inequalities
Laws of Large Numbers
Elec210 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 19
Single Gaussian Random Variable
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 certain value, m,
called the mean.
This rando
Elec210: Lecture 18
Random Vectors
Joint distribution/density/mass functions
Marginal statistics
Conditional densities
Independence and Expectation
http:/www.cs.princeton.edu/~cdecoro/eigenfaces/
Elec210 Lecture 18
1
N Random Variables
An N dimensional ra
Elec210: 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 e
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 k ,
Elec210: Lecture 15
Independence
Expectation of Function of 2 Variables
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, defi
Elec210: Lecture 14
Pairs of continuous random variables
Review of 2D functions, differentiation and integration
Joint cumulative distribution function
Joint probability density function
Elec210 Lecture 14
1
Two Random Variables
One random variable c