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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.

Discrete distribution

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 .

It may be that the set S has elements that are themselves real numbers. In such an instance we could write X ( s ) = 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 = ( 1,2,3,4,5,6 ) . For each s belongs to S , let X ( s ) = 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 ( X = 5 ) = 1 / 6, P ( 2 X 5 ) = 4 / 6, and s belongs to S seem to be reasonable assignments, where ( 2 X 5 ) means ( X = 2,3,4 or 5) and ( X 2 ) means ( X = 1 or 2), in this example.

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    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 ( X = x ) 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 ( x ) = P ( X = x ) , 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 ( X = x ) represents the fraction of times x can be expected to occur. Moreover, to determine the probability associated with the event A R , one would sum the probabilities of the x values in A .

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

  • P ( X = x ) ,
  • x R f ( x ) = 1 ;
  • P ( X A ) = x A f ( x ) , where A R .

Usually let f ( x ) = 0 when x 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 ( x ) = 0 elsewhere, namely, f ( x ) = 0 , x R . Since the probability P ( X = x ) = f ( x ) > 0 when x 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 .

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
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 .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Introduction to statistics. OpenStax CNX. Oct 09, 2007 Download for free at http://cnx.org/content/col10343/1.3
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