# Review of probability theory

 Page 1 / 1
(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 ${}_{XY}=0$ .

Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
Got questions? Join the online conversation and get instant answers!