<< Chapter < Page Chapter >> Page >
(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

Got questions? Get instant answers now!


Probability assignments on intervals a X b

Cumulative distribution
The cumulative distribution function of a random variable X is a function F X such that
F X b X b X 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 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 x F X 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 x x x k x x k F X x

Two random variables defined on an experiment have joint distribution

F X Y a b X a Y b X a Y b

Joint pdf can be obtained if they are jointly continuous

F X Y a b y b x a f X Y x y
( e.g. , f X Y x y x y F X Y x 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 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 and y . For discrete random variables,
p X Y x k y l p X x k p Y y l
for all k and l .


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

g X g X x g x f X x continuous k k g x k p X x k discrete
  • Mean
    X X
  • Second moment
    X 2 X 2
  • Variance
    Var X X X X 2 X 2 X 2
  • Characteristic function
    X u u X
    for u , where 1
  • Correlation between two random variables
    R X Y X Y * y 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 X Y Cov X Y X X Y Y * R X Y X Y *
  • Correlation coefficient
    X Y Cov X Y X Y

Uncorrelated random variables
Two random variables X and Y are uncorrelated if X Y 0 .

Questions & Answers

Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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?
how to fabricate graphene ink ?
for screen printed electrodes ?
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
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Digital communication systems. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10134/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Digital communication systems' conversation and receive update notifications?