# 0.9 Lab 7a - discrete-time random processes (part 1)  (Page 2/5)

 Page 2 / 5

The two most common expectations are the mean ${\mu }_{X}$ and variance ${\sigma }_{X}^{2}$ defined by

${\mu }_{X}=E\left[X\right]={\int }_{-\infty }^{\infty }x{f}_{X}\left(x\right)dx$
${\sigma }_{X}^{2}=E\left[{\left(X-{\mu }_{X}\right)}^{2}\right]={\int }_{-\infty }^{\infty }{\left(x-{\mu }_{X}\right)}^{2}{f}_{X}\left(x\right)dx\phantom{\rule{4pt}{0ex}}.$

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:

${f}_{X}\left(x\right)=\frac{1}{\sqrt{2\pi }{\sigma }_{X}}exp\left(-,\frac{1}{2{\sigma }_{X}^{2}},{\left(x-{\mu }_{X}\right)}^{2}\right)\phantom{\rule{4pt}{0ex}}.$

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 $X\sim N\left(\mu ,{\sigma }^{2}\right)\phantom{\rule{4pt}{0ex}}$ is used to identify $X$ as being Gaussian with mean $\mu$ and variance ${\sigma }^{2}$ .

## 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 $\epsilon$ . We can then characterize the observed value, $X$ , as a random variable, $X=v+\epsilon$ .

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

Suppose we want to use our observation $\left\{{X}_{1},{X}_{2},\cdots ,{X}_{N}\right\}$ 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

${\stackrel{^}{\mu }}_{X}=\frac{1}{N}\sum _{i=1}^{N}{X}_{i}$
${\stackrel{^}{\sigma }}_{X}^{2}=\frac{1}{N-1}\sum _{i=1}^{N}{\left({X}_{i}-{\stackrel{^}{\mu }}_{X}\right)}^{2}\phantom{\rule{4pt}{0ex}}.$

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 ${\stackrel{^}{\mu }}_{X}$ .

$E\left[{\stackrel{^}{\mu }}_{X}\right]=E\left[\frac{1}{N},\sum _{i=1}^{N},{X}_{i}\right]=\frac{1}{N}\sum _{i=1}^{N}E\left[{X}_{i}\right]={\mu }_{X}$
$\begin{array}{ccc}\hfill Var\left[{\stackrel{^}{\mu }}_{X}\right]& =& Var\left[\frac{1}{N},\sum _{i=1}^{N},{X}_{i}\right]=\frac{1}{{N}^{2}}Var\left[\sum _{i=1}^{N},{X}_{i}\right]\hfill \\ & =& \frac{1}{{N}^{2}}\sum _{i=1}^{N}Var\left[{X}_{i}\right]=\frac{{\sigma }_{X}^{2}}{N}\hfill \end{array}$

In both [link] and [link] we have used the i.i.d. assumption. We can also show that $E\left[{\stackrel{^}{\sigma }}_{X}^{2}\right]={\sigma }_{X}^{2}$ .

An estimate $\stackrel{^}{a}$ for some parameter $a$ which has the property $E\left[\stackrel{^}{a}\right]=a$ is said to be an unbiased estimate. An estimator such that $Var\left[\stackrel{^}{a}\right]\to 0$ as $N\to \infty$ 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?

who was the first nanotechnologist
k
Veysel
technologist's thinker father is Richard Feynman but the literature first user scientist Nario Tagunichi.
Veysel
Norio Taniguchi
puvananathan
Interesting
Andr
I need help
Richard
anyone have book of Abdel Salam Hamdy Makhlouf book in pdf Fundamentals of Nanoparticles: Classifications, Synthesis
what happen with The nano material on The deep space.?
It could change the whole space science.
puvananathan
the characteristics of nano materials can be studied by solving which equation?
sibaram
synthesis of nano materials by chemical reaction taking place in aqueous solvents under high temperature and pressure is call?
sibaram
hydrothermal synthesis
ISHFAQ
how can chip be made from sand
is this allso about nanoscale material
Almas
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
STM - Scanning Tunneling Microscope.
puvananathan
Got questions? Join the online conversation and get instant answers!

#### Get Jobilize Job Search Mobile App in your pocket Now! By By Rylee Minllic By Stephen Voron By Jonathan Long By OpenStax By Anh Dao By By OpenStax By Rhodes By CB Biern