Discrete distribution

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This course is a short series of lectures on Introductory Statistics. Topics covered are listed in the Table of Contents. The notes were prepared by EwaPaszek and Marek Kimmel. The development of this course has been supported by NSF 0203396 grant.

Random variable of discrete type

A SAMPLE SPACE S may be difficult to describe if the elements of S are not numbers. Let discuss how one can use a rule by which each simple outcome of a random experiment, an element s of S , may be associated with a real number x .

DEFINITION OF RANDOM VARIABLE
It may be that the set S has elements that are themselves real numbers. In such an instance we could write $X\left(s\right)=s$ so that X is the identity function and the space of X is also S . This is illustrated in the example below.

Let the random experiment be the cast of a die, observing the number of spots on the side facing up. The sample space associated with this experiment is $S=\left(1,2,3,4,5,6\right)$ . For each s belongs to S , let $X\left(s\right)=s$ . The space of the random variable X is then {1,2,3,4,5,6}.

If we associate a probability of 1/6 with each outcome, then, for example, $P\left(X=5\right)=1/6,P\left(2\le X\le 5\right)=4/6,$ and s belongs to S seem to be reasonable assignments, where $\left(2\le X\le 5\right)$ means ( X = 2,3,4 or 5) and $\left(X\le 2\right)$ means ( X = 1 or 2), in this example.

We can recognize two major difficulties:

• In many practical situations the probabilities assigned to the event are unknown.
• Since there are many ways of defining a function X on S , which function do we want to use?

Let X denotes a random variable with one-dimensional space R , a subset of the real numbers. Suppose that the space R contains a countable number of points; that is, R contains either a finite number of points or the points of R can be put into a one-to- one correspondence with the positive integers. Such set R is called a set of discrete points or simply a discrete sample space .

Furthermore, the random variable X is called a random variable of the discrete type , and X is said to have a distribution of the discrete type . For a random variable X of the discrete type, the probability $P\left(X=x\right)$ is frequently denoted by f(x) , and is called the probability density function and it is abbreviated p.d.f. .

Let f(x) be the p.d.f. of the random variable X of the discrete type, and let R be the space of X . Since, $f\left(x\right)=P\left(X=x\right)$ , x belongs to R , f(x) must be positive for x belongs to R and we want all these probabilities to add to 1 because each $P\left(X=x\right)$ represents the fraction of times x can be expected to occur. Moreover, to determine the probability associated with the event $A\subset R$ , one would sum the probabilities of the x values in A .

That is, we want f(x) to satisfy the properties

• $P\left(X=x\right)$ ,
• $\sum _{x\in R}f\left(x\right)=1;$
• $P\left(X\in A\right)=\sum _{x\in A}f\left(x\right)$ , where $A\subset R.$

Usually let $f\left(x\right)=0$ when $x\notin R$ and thus the domain of f(x) is the set of real numbers. When we define the p.d.f. of f(x) and do not say zero elsewhere, then we tacitly mean that f(x) has been defined at all x’s in space R , and it is assumed that $f\left(x\right)=0$ elsewhere, namely, $f\left(x\right)=0$ , $x\notin R$ . Since the probability $P\left(X=x\right)=f\left(x\right)>0$ when $x\in R$ and since R contains all the probabilities associated with X , R is sometimes referred to as the support of X as well as the space of X .

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..
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
characteristics of micro business
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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