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The two most common expectations are the mean μ X and variance σ X 2 defined by

μ X = E [ X ] = - x f X ( x ) d x
σ X 2 = E [ ( X - μ X ) 2 ] = - ( x - μ X ) 2 f X ( x ) d x .

A very important type of random variable is the Gaussian or normal random variable.A Gaussian random variable has a density function of the following form:

f X ( x ) = 1 2 π σ X exp - 1 2 σ X 2 ( x - μ X ) 2 .

Note that a Gaussian random variable is completely characterized by its mean and variance.This is not necessarily the case for other types of distributions. Sometimes, the notation X N ( μ , σ 2 ) is used to identify X as being Gaussian with mean μ and variance σ 2 .

Samples of a random variable

Suppose some random experiment may be characterized by a random variable X whose distribution is unknown. For example, suppose we are measuring a deterministic quantity v , but our measurement is subject to a random measurement error ε . We can then characterize the observed value, X , as a random variable, X = v + ε .

If the distribution of X does not change over time, we may gain further insight into X by making several independent observations { X 1 , X 2 , , X N } . These observations X i , also known as samples , will be independent random variables and have the same distribution F X ( x ) . In this situation, the X i 's are referred to as i.i.d. , for independent and identically distributed . We also sometimes refer to { X 1 , X 2 , , X N } collectively as a sample, or observation, of size N .

Suppose we want to use our observation { X 1 , X 2 , , X N } to estimate the mean and variance of X . Two estimators which should already be familiar to you are the sample mean and sample variance defined by

μ ^ X = 1 N i = 1 N X i
σ ^ X 2 = 1 N - 1 i = 1 N ( X i - μ ^ X ) 2 .

It is important to realize that these sample estimates are functions of random variables, and are therefore themselves random variables.Therefore we can also talk about the statistical properties of the estimators. For example, we can compute the mean and variance of the sample mean μ ^ X .

E μ ^ X = E 1 N i = 1 N X i = 1 N i = 1 N E X i = μ X
V a r μ ^ X = V a r 1 N i = 1 N X i = 1 N 2 V a r i = 1 N X i = 1 N 2 i = 1 N V a r X i = σ X 2 N

In both [link] and [link] we have used the i.i.d. assumption. We can also show that E [ σ ^ X 2 ] = σ X 2 .

An estimate a ^ for some parameter a which has the property E [ a ^ ] = a is said to be an unbiased estimate. An estimator such that V a r [ a ^ ] 0 as N is said to be consistent . These two properties are highly desirable because they imply that if alarge number of samples are used the estimate will be close to the true parameter.

Suppose X is a Gaussian random variable with mean 0 and variance 1. Use the Matlab function random or randn to generate 1000 samples of X , denoted as X 1 , X 2 , ..., X 1000 . See the online help for the random function . Plot them using the Matlab function plot . We will assume our generated samples are i.i.d.

Write Matlab functions to compute the sample mean and sample variance of [link] and [link] without using the predefined mean and var functions. Use these functions to compute the sample meanand sample variance of the samples you just generated.

Inlab report

  1. Submit the plot of samples of X .
  2. Submit the sample mean and the sample variance that you calculated. Why are they not equal to the true mean and true variance?

Questions & Answers

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.
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
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Purdue digital signal processing labs (ece 438). OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10593/1.4
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