# Review of probability theory

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(Blank Abstract)

The focus of this course is on digital communication, which involves transmission of information, in its most general sense,from source to destination using digital technology. Engineering such a system requires modeling both the informationand the transmission media. Interestingly, modeling both digital or analog information and many physical media requires aprobabilistic setting. In this chapter and in the next one we will review the theory of probability, model random signals, andcharacterize their behavior as they traverse through deterministic systems disturbed by noise and interference. Inorder to develop practical models for random phenomena we start with carrying out a random experiment. We then introducedefinitions, rules, and axioms for modeling within the context of the experiment. The outcome of a random experiment isdenoted by  . The sample space  is the set of all possible outcomes of a random experiment. Such outcomescould be an abstract description in words. A scientific experiment should indeed be repeatable where each outcome couldnaturally have an associated probability of occurrence. This is defined formally as the ratio of the number of times the outcomeoccurs to the total number of times the experiment is repeated.

## Random variables

A random variable is the assignment of a real number to each outcome of a random experiment.

Roll a dice. Outcomes $\{{}_{1}, {}_{2}, {}_{3}, {}_{4}, {}_{5}, {}_{6}\}$

${}_{i}$ = $i$ dots on the face of the dice.

$X({}_{i})=i$

## Distributions

Probability assignments on intervals $a< X\le b$

Cumulative distribution
The cumulative distribution function of a random variable $X$ is a function $F(X, (\mathbb{R}, \mathbb{R}))$ such that
$F(X, b)=(X\le b)=(\{\in \colon X()\le b\})$
Continuous Random Variable
A random variable $X$ is continuous if the cumulative distribution function can bewritten in an integral form, or
$F(X, b)=\int_{()} \,d x$ b f X x
and $f(X, x)$ is the probability density function (pdf) ( e.g. , $F(X, x)$ is differentiable and $f(X, x)=\frac{d F(X, x)}{d x}}$ )
Discrete Random Variable
A random variable $X$ is discrete if it only takes at most countably many points( i.e. , $F(X, )$ is piecewise constant). The probability mass function (pmf) is defined as
$p(X, {x}_{k})=(X={x}_{k})=F(X, {x}_{k})-\lim_{x\to {x}_{k}\land (x< {x}_{k})}F(X, x)$

Two random variables defined on an experiment have joint distribution

$F(X, , Y, a, b)=(X\le a, Y\le b)=(\{\in \colon (X()\le a)\land (Y()\le b)\})$

Joint pdf can be obtained if they are jointly continuous

$F(X, , Y, a, b)=\int_{()} \,d y$ b x a f X Y x y
( e.g. , $f(X, , Y, x, y)=\frac{\partial^{2}F(X, , Y, x, y)}{\partial x\partial y}$ )

Joint pmf if they are jointly discrete

$p(X, , Y, {x}_{k}, {y}_{l})=(X={x}_{k}, Y={y}_{l})$

Conditional density function

${f}_{Y|X}(y|x)=\frac{f(X, , Y, x, y)}{f(X, x)}$
for all $x$ with $f(X, x)> 0$ otherwise conditional density is not defined for those valuesof $x$ with $f(X, x)=0$

Two random variables are independent if

$f(X, , Y, x, y)=f(X, x)f(Y, y)$
for all $x\in \mathbb{R}$ and $y\in \mathbb{R}$ . For discrete random variables,
$p(X, , Y, {x}_{k}, {y}_{l})=p(X, {x}_{k})p(Y, {y}_{l})$
for all $k$ and $l$ .

## Moments

Statistical quantities to represent some of the characteristics of a random variable.

$\langle g(X)\rangle =(g(X))=\begin{cases}\int_{()} \,d x & \text{if }\end{cases}$ g x f X x continuous k k g x k p X x k discrete
• Mean
${}_{X}=\langle X\rangle$
• Second moment
$(X^{2})=\langle X^{2}\rangle$
• Variance
$\mathrm{Var}(X)=\sigma(X)^2=\langle (X-{}_{X})^{2}\rangle =\langle X^{2}\rangle -{}_{X}^{2}$
• Characteristic function
${}_{X}(u)=\langle e^{iuX}\rangle$
for $u\in \mathbb{R}$ , where $i=\sqrt{-1}$
• Correlation between two random variables
${R}_{XY}=\langle X{Y}^{*}\rangle =\begin{cases}\int_{()} \,d y & \text{if }\end{cases}$ x x y * f X Y x y X and Y are jointly continuous k k l l x k y l * p X Y x k y l X and Y are jointly discrete
• Covariance
${C}_{XY}=\mathrm{Cov}(X, Y)=\langle (X-{}_{X})(Y-{}_{Y})^{*}\rangle ={R}_{XY}-{}_{X}{}_{Y}^{*}$
• Correlation coefficient
${}_{XY}=\frac{\mathrm{Cov}(X, Y)}{{}_{X}{}_{Y}}$

Uncorrelated random variables
Two random variables $X$ and $Y$ are uncorrelated if ${}_{XY}=0$ .

where we get a research paper on Nano chemistry....?
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
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
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
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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