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Second-order description

Practical and incomplete statistics

The mean function of a random process X t is defined as the expected value of X t for all t 's.
μ X t X t x x f X t x continuous k x k p X t x k discrete
The autocorrelation function of the random process X t is defined as
R X t 2 t 1 X t 2 X t 1 x 2 x 1 x 2 x 1 f X t 2 X t 1 x 2 x 1 continuous k l x l x k p X t 2 X t 1 x l x k discrete


If X t is second-order stationary, then R X t 2 t 1 only depends on t 2 t 1 .

R X t 2 t 1 X t 2 X t 1 x 1 x 2 x 2 x 1 f X t 2 X t 1 x 2 x 1
R X t 2 t 1 x 1 x 2 x 2 x 1 f X t 2 - t 1 X 0 x 2 x 1 R X t 2 t 1 0

If R X t 2 t 1 depends on t 2 t 1 only, then we will represent the autocorrelation with only one variable τ t 2 t 1

R X τ R X t 2 t 1 R X t 2 t 1


  1. R X 0 0
  2. R X τ R X τ
  3. R X τ R X 0

X t 2 f 0 t Θ ω and Θ is uniformly distributed between 0 and 2 . The mean function

μ X t X t 2 f 0 t Θ θ 0 2 2 f 0 t θ 1 2 0

The autocorrelation function

R X t τ t X t + τ X t 2 f 0 t τ Θ 2 f 0 t Θ 1 2 2 f 0 τ 1 2 2 f 0 2 t τ 2 Θ 1 2 2 f 0 τ 1 2 θ 0 2 2 f 0 2 t τ 2 θ 1 2 1 2 2 f 0 τ
Not a function of t since the second term in the right hand side of the equality in [link] is zero.

Toss a fair coin every T seconds. Since X t is a discrete valued random process, the statistical characteristics can be captured by the pmf and the mean function is written as

μ X t X t 1 2 -1 1 2 1 0
R X t 2 t 1 k k l l x k x l p X t 2 X t 1 x k x l 1 1 1 2 -1 -1 1 2 1
when n T t 1 n 1 T and n T t 2 n 1 T
R X t 2 t 1 1 1 1 4 -1 -1 1 4 -1 1 1 4 1 -1 1 4 0
when n T t 1 n 1 T and m T t 2 m 1 T with n m
R X t 2 t 1 1 n T t 1 n 1 T n T t 2 n 1 T 0
A function of t 1 and t 2 .

Wide Sense Stationary
A process is said to be wide sense stationary if μ X is constant and R X t 2 t 1 is only a function of t 2 t 1 .

If X t is strictly stationary, then it is wide sense stationary. The converse is not necessarily true.

Autocovariance of a random process is defined as
C X t 2 t 1 X t 2 μ X t 2 X t 1 μ X t 1 R X t 2 t 1 μ X t 2 μ X t 1

The variance of X t is Var X t C X t t

Two processes defined on one experiment ( [link] ).

The crosscorrelation function of a pair of random processes is defined as
R X Y t 2 t 1 X t 2 Y t 1 y x x y f X t 2 Y t 1 x y
C X Y t 2 t 1 R X Y t 2 t 1 μ X t 2 μ Y t 1
Jointly Wide Sense Stationary
The random processes X t and Y t are said to be jointly wide sense stationary if R X Y t 2 t 1 is a function of t 2 t 1 only and μ X t and μ Y t are constant.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Principles of digital communications. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10805/1.1
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