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In order to study the characteristics of a random process , let us look at some of the basic properties and operations of a random process. Below wewill focus on the operations of the random signals that compose our random processes. We will denote our random process with X and a random variable from a random process or signal by x .

Mean value

Finding the average value of a set of random signals or random variables is probably the most fundamental concepts we use inevaluating random processes through any sort of statistical method. The mean of a random process is the average of all realizations of that process. In order to find this average, we must look at a random signal over arange of time (possible values) and determine our average from this set of values. The mean , or average, of a random process, x t , is given by the following equation:

m x t x t X X x x f x
This equation may seem quite cluttered at first glance, but we want to introduce you to the various notations used torepresent the mean of a random signal or process. Throughout texts and other readings, remember that these will all equalthe same thing. The symbol, x t , and the X with a bar over it are often used as a short-hand to represent anaverage, so you might see it in certain textbooks. The other important notation used is, X , which represents the "expected value of X " or the mathematical expectation. This notation is very common and will appearagain.

If the random variables, which make up our random process, are discrete or quantized values, such as in a binary process,then the integrals become summations over all the possible values of the random variable. In this case, our expectedvalue becomes

x n x x n
If we have two random signals or variables, their averages can reveal how the two signals interact. If the product of the two individualaverages of both signals do not equal the average of the product of the two signals, then the twosignals are said to be linearly independent , also referred to as uncorrelated .

In the case where we have a random process in which only one sample can be viewed at a time, then we will often not haveall the information available to calculate the mean using the density function as shown above. In this case we mustestimate the mean through the time-average mean , discussed later. For fields such as signal processing that deal mainly withdiscrete signals and values, then these are the averages most commonly used.

Properties of the mean

  • The expected value of a constant, , is the constant:
  • Adding a constant, , to each term increases the expected value by that constant:
    X X
  • Multiplying the random variable by a constant, , multiplies the expected value by that constant.
    X X
  • The expected value of the sum of two or more random variables, is the sum of each individual expectedvalue.
    X Y X Y

Mean-square value

If we look at the second moment of the term (we now look at x 2 in the integral), then we will have the mean-square value of our random process. As you would expect, this is written as

X 2 X 2 x x 2 f x
This equation is also often referred to as the average power of a process or signal.


Now that we have an idea about the average value or values that a random process takes, we are often interested in seeingjust how spread out the different random values might be. To do this, we look at the variance which is a measure of this spread. The variance, often denoted by 2 , is written as follows:

2 Var X X X 2 x x X 2 f x
Using the rules for the expected value, we can rewrite this formula as the following form, which is commonly seen:
2 X 2 X 2 X 2 X 2

Standard deviation

Another common statistical tool is the standard deviation. Once you know how to calculate the variance, the standarddeviation is simply the square root of the variance , or .

Properties of variance

  • The variance of a constant, , equals zero:
    Var 0
  • Adding a constant, , to a random variable does not affect the variance because the mean increases by the same value:
    Var X X X
  • Multiplying the random variable by a constant, , increases the variance by the square of the constant:
    Var X X 2 X
  • The variance of the sum of two random variables only equals the sum of the variances if the variable are independent .
    Var X Y X Y X Y
    Otherwise, if the random variable are not independent, then we must also include the covariance of the product of the variablesas follows:
    Var X Y X 2 Cov X Y Y

Time averages

In the case where we can not view the entire ensemble of the random process, we must use time averages to estimate thevalues of the mean and variance for the process. Generally, this will only give us acceptable results for independent and ergodic processes, meaning those processes in which each signal or member of the process seems to have thesame statistical behavior as the entire process. The time averages will also only be taken over a finite interval sincewe will only be able to see a finite part of the sample.

Estimating the mean

For the ergodic random process, x t , we will estimate the mean using the time averaging function defined as

X X 1 T t 0 T X t
However, for most real-world situations we will be dealing with discrete values in our computations and signals. Wewill represent this mean as
X X 1 N n 1 N X n

Estimating the variance

Once the mean of our random process has been estimated then we can simply use those values in the following varianceequation (introduced in one of the above sections)

x 2 X 2 X 2


Let us now look at how some of the formulas and concepts above apply to a simple example. We will just look at a single,continuous random variable for this example, but the calculations and methods are the same for a random process.For this example, we will consider a random variable having the probability density function described below and shown in .

f x 1 10 10 x 20 0

Probability density function

A uniform probability density function.

First, we will use to solve for the mean value.

X x 10 20 x 1 10 x 10 20 1 10 x 2 2 1 10 200 50 15
Using we can obtain the mean-square value for the above density function.
X 2 x 10 20 x 2 1 10 x 10 20 1 10 x 3 3 1 10 8000 3 1000 3 233.33
And finally, let us solve for the variance of this function.
2 X 2 X 2 233.33 15 2 8.33

Questions & Answers

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
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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.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
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Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
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Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Intro to digital signal processing. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10203/1.4
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