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We formalize as follows:

A pair { X , Y } of random variables considered jointly is treated as the pair of coordinate functions for a two-dimensional random vector W = ( X , Y ) . To each ω Ω , W assigns the pair of real numbers ( t , u ) , where X ( ω ) = t and Y ( ω ) = u . If we represent the pair of values { t , u } as the point ( t , u ) on the plane, then W ( ω ) = ( t , u ) , so that

W = ( X , Y ) : Ω R 2

is a mapping from the basic space Ω to the plane R 2 . Since W is a function, all mapping ideas extend. The inverse mapping W - 1 plays a role analogous to that of the inverse mapping X - 1 for a real random variable. A two-dimensional vector W is a random vector iff W - 1 ( Q ) is an event for each reasonable set (technically, each Borel set) on the plane.

A fundamental result from measure theory ensures

W = ( X , Y ) is a random vector iff each of the coordinate functions X and Y is a random variable.

In the selection example above, we model X (the number of juniors selected)   and Y (the number of seniors selected) as random variables. Hence the vector-valued function

Induced distribution and the joint distribution function

In a manner parallel to that for the single-variable case, we obtain a mapping of probability mass from the basic space to the plane. Since W - 1 ( Q ) is an event for each reasonable set Q on the plane, we may assign to Q the probability mass

P X Y ( Q ) = P [ W - 1 ( Q ) ] = P [ ( X , Y ) - 1 ( Q ) ]

Because of the preservation of set operations by inverse mappings as in the single-variable case, the mass assignment determines P X Y as a probability measure on the subsets of the plane R 2 . The argument parallels that for the single-variable case. The result is the probability distribution induced by W = ( X , Y ) . To determine the probability that the vector-valued function W = ( X , Y ) takes on a (vector) value in region Q , we simply determine how much induced probability mass is in that region.

Induced distribution and probability calculations

To determine P ( 1 X 3 , Y > 0 ) , we determine the region for which the first coordinate value (which we call t ) is between one and three and the second coordinate value (which we call u ) is greater than zero. This corresponds to the set Q of points on the plane with 1 t 3 and u > 0 . Gometrically, this is the strip on the plane bounded by (but not including) the horizontal axis and by thevertical lines t = 1 and t = 3 (included). The problem is to determine how much probability mass lies in that strip. How this is acheived depends upon the nature of thedistribution and how it is described.

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As in the single-variable case, we have a distribution function.


The joint distribution function F X Y for W = ( X , Y ) is given by

F X Y ( t , u ) = P ( X t , Y u ) ( t , u ) R 2

This means that F X Y ( t , u ) is equal to the probability mass in the region Q t u on the plane such that the first coordinate is less than or equal to t and the second coordinate is less than or equal to u . Formally, we may write

F X Y ( t , u ) = P [ ( X , Y ) Q t u ] , where Q t u = { ( r , s ) : r t , s u }

Now for a given point ( a , b ) , the region Q a b is the set of points ( t , u ) on the plane which are on or to the left of the vertical line through ( t , 0 ) and on or below the horizontal line through ( 0 , u ) (see Figure 1 for specific point t = a , u = b ). We refer to such regions as semiinfinite intervals on the plane.

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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
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
Samson Reply

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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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