# Random vectors and joint distributions  (Page 2/3)

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We formalize as follows:

A pair $\left\{X,\phantom{\rule{0.166667em}{0ex}}Y\right\}$ of random variables considered jointly is treated as the pair of coordinate functions for a two-dimensional random vector $W=\left(X,\phantom{\rule{0.166667em}{0ex}}Y\right)$ . To each $\omega \in \Omega$ , W assigns the pair of real numbers $\left(t,\phantom{\rule{0.166667em}{0ex}}u\right)$ , where $X\left(\omega \right)=t$ and $Y\left(\omega \right)=u$ . If we represent the pair of values $\left\{t,\phantom{\rule{0.166667em}{0ex}}u\right\}$ as the point $\left(t,\phantom{\rule{0.166667em}{0ex}}u\right)$ on the plane, then $W\left(\omega \right)=\left(t,\phantom{\rule{0.166667em}{0ex}}u\right)$ , so that

$W=\left(X,\phantom{\rule{0.166667em}{0ex}}Y\right):\phantom{\rule{0.277778em}{0ex}}\Omega \to {\mathbf{R}}^{2}$

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 ${W}^{-1}\left(Q\right)$ is an event for each reasonable set (technically, each Borel set) on the plane.

A fundamental result from measure theory ensures

$W=\left(X,\phantom{\rule{0.166667em}{0ex}}Y\right)$ is a random vector iff each of the coordinate functions X and Y is a random variable.

In the selection example above, we model $X\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}$ (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 ${W}^{-1}\left(Q\right)$ is an event for each reasonable set Q on the plane, we may assign to Q the probability mass

${P}_{XY}\left(Q\right)=P\left[{W}^{-1}\left(Q\right)\right]=P\left[{\left(X,\phantom{\rule{0.166667em}{0ex}}Y\right)}^{-1}\left(Q\right)\right]$

Because of the preservation of set operations by inverse mappings as in the single-variable case, the mass assignment determines ${P}_{XY}$ 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 $W=\left(X,\phantom{\rule{0.166667em}{0ex}}Y\right)$ . To determine the probability that the vector-valued function $W=\left(X,\phantom{\rule{0.166667em}{0ex}}Y\right)$ 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 $P\left(1\le X\le 3,\phantom{\rule{0.277778em}{0ex}}Y>0\right)$ , 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 $1\le t\le 3$ 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.

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

Definition

The joint distribution function ${F}_{XY}$ for $W=\left(X,\phantom{\rule{0.166667em}{0ex}}Y\right)$ is given by

${F}_{XY}\left(t,\phantom{\rule{0.166667em}{0ex}}u\right)=P\left(X\le t,\phantom{\rule{0.277778em}{0ex}}Y\le u\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall \phantom{\rule{0.277778em}{0ex}}\left(t,\phantom{\rule{0.166667em}{0ex}}u\right)\in {\mathbf{R}}^{2}$

This means that ${F}_{XY}\left(t,\phantom{\rule{0.166667em}{0ex}}u\right)$ is equal to the probability mass in the region ${Q}_{tu}$ 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

${F}_{XY}\left(t,\phantom{\rule{0.166667em}{0ex}}u\right)=P\left[\left(X,\phantom{\rule{0.166667em}{0ex}}Y\right)\in {Q}_{tu}\right],\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\text{where}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{Q}_{tu}=\left\{\left(r,\phantom{\rule{0.166667em}{0ex}}s\right):\phantom{\rule{0.166667em}{0ex}}r\le t,\phantom{\rule{0.166667em}{0ex}}s\le u\right\}$

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

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
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..
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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
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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
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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
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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
A fair die is tossed 180 times. Find the probability P that the face 6 will appear between 29 and 32 times inclusive