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

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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?

how do they get the third part x = (32)5/4
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
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Sherica
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Sherica
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Tamia
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Uday
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a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
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is it 3×y ?
J, combine like terms 7x-4y
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what's the easiest and fastest way to the synthesize AgNP?
China
Cied
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I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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Yasmin
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Cesar
I'm interested in nanotube
Uday
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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Stotaw
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Azam
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Prasenjit
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Azam
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Prasenjit
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Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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