# 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 ${}_{XY}=0$ .

With the aid of diagrams,compare the short run equilibrium positions of a perfect competitor and an imperfect competitors
What is the term consultation in economics?
why is it that the long run Average cost curve does not touch that of the short run curve at its minimum?
In few words, what is the role of interest rate in economy?
what is population
total number of people in a given area or country
Clement
the total number of people at a given area or country
Clement
total number of people in a given area or country
okhiria
What is price Elasticity of demand?.
Samuel
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.
EMMANUEL
why is it that the long run Average cost doesn't touch the short run cost curve at its minimum?
Baah
what is supply
Precious
please rice and beans will be what type of demand but note we mostly cook it together
complementary or joint demand
Yillah
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.
Carlos
what are the money value
Nothing more than a purchase power, in other words, \$100 now, must have the same value after 1 year.
Carlos
what is Monopoly
what is money
It can be define as a big transaction that can control any business for one place to another base.
Akinlo
money is recognisable note to accept both parties selling and buying
Hassan
i don still understan
Rene
hey
Abdul
hi
Rene
money is anything generally accepted as a medium of exchange
Awwal
Money is anything generally accepted as a medium of exchange and for the settlement of goods and services .
Korda
hi good ppl, pls help out
Tumi
discuss human and natural resources as develop strategies ro improving living condition of citizens in developing countries.
Tumi
I don't understand the question.
Naomi
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?
Naomi
money is a legal thunder generally accepted as a medium of exchange for the payment of debt ,goods and services
Naomi
money is a way of payment.
Carlos
money is any thing that is generally accepted as a medium of exchange good for good and settlement of debt and means of payment
Yillah
money is nothing but a object which is used for exchange of goods and services.
Harshita
money is anything that is generally accepted as payment of goods and services and settlement of debt
Rebecca
what is demand
demand is where the customer is willing and able to buy goods and services during a given period of time
idk
demand is the ability and willingness of an individual to buy goods and services at a given price in a particular period of time
Alpha
demand is the ability to buy a specific quantities of goods and services at a given price and at a specific period of time
rosemary
what are the rules of demand
rosemary
Rosemary Nsebon, Do you mean laws of demand?
Alpha
what are the rules of demand
Rene
the rule of demand is the higher the price the lower the quantity demanded and the lower the price the higher the quantity demanded
mbi
thank
Rene
what is unemployment
Rebecca
unemployment is a scenario or a phenomenon in an economy whereby people are willing are able to work but cannot a job
mbi
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
Enables individuals and countries to consume a variety of goods and services
Iddrisu
what is the meaning of competency
competency is an ability and courage to do something perfectly
Abdullahi
Segun
rival
Ray
thanks 🙏 it is also the same with the core competency
A sufficient supply
Ebenezer
Ebenezer you mean the (core competency) right?
what is mean,median and mode
mean is the average number of a given data
Gallant
median is the middle number of a given data
Gallant
in a given data sorry
Mitchel
hi
Sajib
Pls am new here
Physcal
what are development bank in Nigeria
.hi
Physcal
hi
hw
Physcal
and cool
Rosie
nice to meet everyone
Rosie
hi how are dears
Mumtaz
how can we development economic in our country
Mumtaz
hi
Charm
Payroll and​ 4p
Wasuroj
Agriculture
Wasuroj
Export
Wasuroj
Transport
Wasuroj
Change management​ and​ cerrancy
Wasuroj
Empoyee
Wasuroj
Lawyer army and​ Lawyer​
Wasuroj
animal husbandry essay
what's the primary location of capital and money market respectively
ALIMI
Difference between extinct and extici spicies
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.
The formulation of new memories is sometimes called ________, and the process of bringing up old memories is called ________.
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