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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

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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

With the aid of diagrams,compare the short run equilibrium positions of a perfect competitor and an imperfect competitors
Monlay Reply
What is the term consultation in economics?
Nthabiseng Reply
why is it that the long run Average cost curve does not touch that of the short run curve at its minimum?
Baah Reply
In few words, what is the role of interest rate in economy?
Carlos Reply
what is population
Nyakeh Reply
total number of people in a given area or country
the total number of people at a given area or country
total number of people in a given area or country
What is price Elasticity of demand?.
The responsiveness of the quantity of a commodity demanded to a change in its price, expressed as the percentage change in quantity demanded divided by the percentage change in price.
why is it that the long run Average cost doesn't touch the short run cost curve at its minimum?
what is supply
please rice and beans will be what type of demand but note we mostly cook it together
Oladosu Reply
complementary or joint demand
but did you know you can demand for rice without beans so how is it joint or complementary
From my point of view, rice and beans are replaceable goods, hence they can't be complementary.
what are the money value
Wisdom Reply
Nothing more than a purchase power, in other words, $100 now, must have the same value after 1 year.
what is Monopoly
Rebecca Reply
what is money
Lawal Reply
It can be define as a big transaction that can control any business for one place to another base.
money is recognisable note to accept both parties selling and buying
i don still understan
money is anything generally accepted as a medium of exchange
Money is anything generally accepted as a medium of exchange and for the settlement of goods and services .
hi good ppl, pls help out
discuss human and natural resources as develop strategies ro improving living condition of citizens in developing countries.
I don't understand the question.
it's a form of currency used for 2 or more individuals or parties in order to reach their amicable personal or business attainment. one must understand that money itself can manifest in multiple fashions for which the individuals or parties adheres.
are u trying to say we shld discuss ways in which human natural resources help in improving living condition of citizens in developing countries?
money is a legal thunder generally accepted as a medium of exchange for the payment of debt ,goods and services
money is a way of payment.
money is any thing that is generally accepted as a medium of exchange good for good and settlement of debt and means of payment
money is nothing but a object which is used for exchange of goods and services.
money is anything that is generally accepted as payment of goods and services and settlement of debt
what is demand
Melissa Reply
demand is where the customer is willing and able to buy goods and services during a given period of time
demand is the ability and willingness of an individual to buy goods and services at a given price in a particular period of time
demand is the ability to buy a specific quantities of goods and services at a given price and at a specific period of time
what are the rules of demand
Rosemary Nsebon, Do you mean laws of demand?
what are the rules of demand
the rule of demand is the higher the price the lower the quantity demanded and the lower the price the higher the quantity demanded
what is unemployment
unemployment is a scenario or a phenomenon in an economy whereby people are willing are able to work but cannot a job
Suppose you have a team of two workers: one is a baker and one is a chef. Explain why the kitchen can produce more meals in a given period of time if each worker specializes in what they do best than if each worker tries to do everything from appetizer to dessert. please I need a urgent answer
Oladosu Reply
Enables individuals and countries to consume a variety of goods and services
what is the meaning of competency
Oladosu Reply
competency is an ability and courage to do something perfectly
ability to perform some task
thanks 🙏 it is also the same with the core competency
A sufficient supply
Ebenezer you mean the (core competency) right?
what is mean,median and mode
Ikeh Reply
mean is the average number of a given data
median is the middle number of a given data
in a given data sorry
Pls am new here
what are development bank in Nigeria
Adedigba Reply
and cool
nice to meet everyone
hi how are dears
how can we development economic in our country
Payroll and​ 4p
Change management​ and​ cerrancy
Lawyer army and​ Lawyer​
animal husbandry essay
Rakuane Reply
what's the primary location of capital and money market respectively
Difference between extinct and extici spicies
Amanpreet Reply
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
Beverly Reply
Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.
Mapo Reply
The formulation of new memories is sometimes called ________, and the process of bringing up old memories is called ________.
Mapo Reply
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Source:  OpenStax, Digital communication systems. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10134/1.3
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