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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

how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
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Do somebody tell me a best nano engineering book for beginners?
s. Reply
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Devang Reply
are you nano engineer ?
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.
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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?
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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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
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I'm interested in nanotube
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Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
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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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