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The concept of independence for classes of events is developed in terms of a product rule. Recall that for a real random variable X, the inverse image of each reasonable subset M on the real line (i.e., the set of all outcomes which are mapped into M by X) is an event. Similarly, the inverse image of N by random variable Y is an event. We extend the notion of independence to a pair of random variables by requiring independence of the events they determine in this fashion. This condition may be stated in terms of the product rule P(X in M, Y in N) = P(X in M)P(Y in N) for all Borel sets M, N.This product rule holds for the distribution functions FXY(t,u) = FX(t)FY(u) for all t, u. And similarly for density functions when they exist. This condition puts restrictions on the nature of the probability mass distribution on the plane. For a rectangle with sides M, Nthe probability mass in M x N is P(X in M)P(Y in N). Extension to general classes is simple and immediate.

Introduction

The concept of independence for classes of events is developed in terms of a product rule. In this unit, we extend the concept to classes of random variables.

Independent pairs

Recall that for a random variable X , the inverse image X - 1 ( M ) (i.e., the set of all outcomes ω Ω which are mapped into M by X ) is an event for each reasonable subset M on the real line. Similarly, the inverse image Y - 1 ( N ) is an event determined by random variable Y for each reasonable set N . We extend the notion of independence to a pair of random variables by requiring independence of the events they determine. More precisely,

Definition

A pair { X , Y } of random variables is (stochastically) independent iff each pair of events { X - 1 ( M ) , Y - 1 ( N ) } is independent.

This condition may be stated in terms of the product rule

P ( X M , Y N ) = P ( X M ) P ( Y N ) for all (Borel) sets M , N

Independence implies

F X Y ( t , u ) = P ( X ( - , t ] , Y ( - , u ] ) = P ( X ( - , t ] ) P ( Y ( - , u ] ) =
F X ( t ) F Y ( u ) t , u

Note that the product rule on the distribution function is equivalent to the condition the product rule holds for the inverse images of a special class of sets { M , N } of the form M = ( - , t ] and N = ( - , u ] . An important theorem from measure theory ensures that if the product rule holds for this special classit holds for the general class of { M , N } . Thus we may assert

The pair { X , Y } is independent iff the following product rule holds

F X Y ( t , u ) = F X ( t ) F Y ( u ) t , u

An independent pair

Suppose F X Y ( t , u ) = 1 - e - α t 1 - e - β u 0 t , 0 u . Taking limits shows

F X ( t ) = lim u F X Y ( t , u ) = 1 - e - α t and F Y ( u ) = lim t F X Y ( t , u ) = 1 - e - β u

so that the product rule F X Y ( t , u ) = F X ( t ) F Y ( u ) holds. The pair { X , Y } is therefore independent.

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If there is a joint density function, then the relationship to the joint distribution function makes it clear that the pair is independent iff the product rule holds for the density. That is, thepair is independent iff

f X Y ( t , u ) = f X ( t ) f Y ( u ) t , u

Joint uniform distribution on a rectangle

Suppose the joint probability mass distributions induced by the pair { X , Y } is uniform on a rectangle with sides I 1 = [ a , b ] and I 2 = [ c , d ] . Since the area is ( b - a ) ( d - c ) , the constant value of f X Y is 1 / ( b - a ) ( d - c ) . Simple integration gives

f X ( t ) = 1 ( b - a ) ( d - c ) c d d u = 1 b - a a t b and
f Y ( u ) = 1 ( b - a ) ( d - c ) a b d t = 1 d - c c u d

Thus it follows that X is uniform on [ a , b ] , Y is uniform on [ c , d ] , and f X Y ( t , u ) = f X ( t ) f Y ( u ) for all t , u , so that the pair { X , Y } is independent. The converse is also true: if the pair is independent with X uniform on [ a , b ] and Y is uniform on [ c , d ] , the the pair has uniform joint distribution on I 1 × I 2 .

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Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
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
Google
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
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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
Smarajit Reply
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A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive
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Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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