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Two (or more) random variables can be defined over the same sample space. Just as with jointly defined events, the joint distribution function is easily defined.

P X Y x y X x Y y
The joint probability density function p X Y x y is related to the distribution function via double integration.
P X Y x y x y p X Y
or p X Y x y x y P X Y x y Since y P X Y x y P X x , the so-called marginal density functions can be related to the joint density function.
p X x p X Y x
and p Y y p X Y y

Extending the ideas of conditional probabilities, the conditional probability density function p X | Y x | Y = y is defined (when p Y y 0 ) as

p X | Y x | Y = y p X Y x y p Y y
Two random variables are statistically independent when p X | Y x | Y = y p X x , which is equivalent to the condition that the joint density function is separable: p X Y x y p X x p Y y .

For jointly defined random variables, expected values are defined similarly as with single random variables. Probably themost important joint moment is the covariance :

cov X Y X Y X Y
where X Y y x x y p X Y x y Related to the covariance is the (confusingly named) correlation coefficient : the covariance normalized by the standard deviations of the component random variables. p X , Y cov X Y X Y When two random variables are uncorrelated , their covariance and correlation coefficient equals zero so that X Y X Y . Statistically independent random variables are always uncorrelated, but uncorrelated random variables can bedependent.
Let X be uniformly distributed over -1 1 and let Y X 2 . The two random variables are uncorrelated, but are clearly not independent.

A conditional expected value is the mean of the conditional density.

Y X x p X | Y x | Y = y
Note that the conditional expected value is now a function of Y and is therefore a random variable. Consequently, it too has an expected value, which is easily evaluated to be the expected value of X .
Y X y x x p X | Y x | Y = y p Y y X
More generally, the expected value of a function of two random variables can be shown to be the expected value of a conditionalexpected value: f X Y Y f X Y . This kind of calculation is frequently simpler to evaluate than trying to find the expected value of f X Y "all at once." A particularly interesting example of this simplicity is the random sum of random variables . Let L be a random variable and X l a sequence of random variables. We will find occasion to consider the quantity l 1 L X l . Assuming that each component of the sequence has the same expected value X , the expected value of the sum is found to be
S L L l 1 L X l L X L X

Questions & Answers

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
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 .
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.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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 did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Statistical signal processing. OpenStax CNX. Dec 05, 2011 Download for free at http://cnx.org/content/col11382/1.1
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