# 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
$({S}_{L})=((L, \sum_{l=1}^{L} {X}_{l}))=(L(X))=(L)(X)$

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
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