# 1.2 Jointly distributed random variables

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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.

$P(X, , Y, x, y)\equiv (\{X\le x\}\cap \{Y\le y\})$
The joint probability density function $p(X, , Y, x, y)$ is related to the distribution function via double integration.
$P(X, , Y, x, y)=\int_{()} \,d$ x y p X Y
or $p(X, , Y, x, y)=\frac{\partial^{2}P(X, , Y, x, y)}{\partial x\partial y}$ Since $\lim_{y\to }y\to$ P X Y x y P X x , the so-called marginal density functions can be related to the joint density function.
$p(X, x)=\int_{()} \,d$ p X Y x
and $p(Y, y)=\int_{()} \,d$ p X Y y

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

${p}_{X|Y}(x|Y=y)=\frac{p(X, , Y, x, y)}{p(Y, y)}$
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 :

$\mathrm{cov}(X, Y)\equiv (XY)-(X)(Y)$
where $(XY)=\int_{()} \,d y$ 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. ${p}_{X,Y}=\frac{\mathrm{cov}(X, Y)}{{}_{X}{}_{Y}}$ When two random variables are uncorrelated , their covariance and correlation coefficient equals zero so that $(XY)=(X)(Y)$ . Statistically independent random variables are always uncorrelated, but uncorrelated random variables can bedependent.
Let $X$ be uniformly distributed over $\left[-1 , 1\right]$ 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.

$(Y, X)=\int_{()} \,d x$ 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 $X$ .
$((Y, X))=\int_{()} \,d y$ 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 $\sum_{l=1}^{L} {X}_{l}$ . Assuming that each component of the sequence has the same expected value $(X)$ , the expected value of the sum is found to be
$({S}_{L})=((L, \sum_{l=1}^{L} {X}_{l}))=(L(X))=(L)(X)$

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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?
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
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