SciTech-Mathmatics-Probability and Statistics: Differencing "mind"/"language"/"Concept"/"ideal"/"Context"/"notation"/"Symbol"/"Term"/"Axiom"/"Definition"/&

SciTech-Mathmatics-Probability and Statistics: Differencing:

"mind"/"language"/"Concept"/"ideal"/"Context"/"notation"/"Term"/"Axiom"/"Definition"/"Condition"/"Property"/"Law"

occur/ happen / obtain / get ?

Example:

the class of Problem

• Problem->Evidence-> Recognition -> Evaluation -> Study ->
  
  considerations,choices,decisions,conclusions, attempt/try,effort, guess, adopt, reach, demonstrate,
  
  Express,Consider/Consideration,interpret
  
  chance/choice: choose / select
  
  decision: determine /
  
  assignment: assign to
• The theories and techniques that will be prensented in this book,
  
  have served as valuable guides and tools, in almost all aspects of the design and analysis of effective experimentation.

EXAMPLE:

• "The subjective nature of science, is also revealed in the actual problem that a particular scientist chooses to study from the class of problems that might have been chosen,"
• in the experiments that are selected in carrying out this study, and in the conclusions drawn from the experimental data."
• "The mathmatical theory of probability and Statistics can plays an important part in these choices,decisions, and conclusions."

• Almost all work in the mathmatical theory of probability, from most elementary textbooks to most advanced research,are related to following two problems:

  • Methods for determining the probabilitis of certain events from the specified probabilities of each possible outcome of an experiment;
  • Methods for revising the probabilitis of each possible events when additional relevant information is obtained.
  • These methods are based on standard mathmatical techniques.

• the Sample Space of an experiment can be thought of as a set, or collection,
  
  of different possible outcomes;

• and Each Outcome can be thought of as a Point, or an Element, in the Sample space.

• similarly Events can be thought of as Subsets of the Sample Space.

• A probability measure, or simply a probability, on a sample space S, is a specification of numbers Pr(A) for all events A that satisfy Axioms 1,2 and 3:

  • Axiom I: the first axiom states that the probability of every event MUST BE nonnegative.
    
    For every event A, Pr(A) ≥ 0.
  • Axiom II: the second axiom states that if an event is certain to occur, the probability of that event is 1.
    
    Pr(S) = 1.
  • Axiom III: "these considerations" lead to the third axiom:
    
    For every infinite sequence of disjoint events, \(\large \begin{array}{*} A_1, A_2, \cdots,\ Pr\left( \underset{i=1}{ \overset{\infty}{ \bigcup}} { A_{i} } \right) = \underset{i=1}{ \overset{\infty}{ \sum}} { Pr(A_{i}) } \\
    \end{array}\)

• Properties of The Probability Theory:

  • Closure Property, for every event/set, is subsets of the Sample Space S.
  • Identity(Certainty of 1) property $\large Pr(S) = 1 $, if an event is CERTAIN to occur, then the PROBABILITY of that is 1.lllllll
  • Zero(Certainty of 0) property: $\large Pr(\emptyset) = 0 $,
  • Nonnegative property: $\large Pr(A) \geq 0 $
  • Additive property: $\large Pr( A \bigcup B ) = Pr(A) + Pr(B) $
  • Commutative property:
    
    $ A \bigcup B = B \bigcup A,\ A \bigcap B = B \bigcap A $
  • Associative property:
    
    $ (A \bigcup B) \bigcup C = A \bigcup ( B \bigcup C ),\ (A \bigcap B) \bigcap C = A \bigcap ( B \bigcap C ) $
  • Distributive property:
    
    $ A \bigcup ( B \bigcap C) = (A \bigcup B) \bigcap (A \bigcup C),\ A \bigcap ( B \bigcup C) = (A \bigcup B) \bigcap (A \bigcup C) $

"Because we can interpret:

• Outcomes as Events of a set,
• Events as Subsets of a set,

the Language and Concepts of set theory, provide a natural context for the development of probability theory

the basic ideals and notation of set theory will now be reviewed."

Education / Training

University/College/Institute/Academy

Major

Subject

"Probability and Statistics" is a subject for the students of mathmatics major.

"Design and Analysis of Experiments"

Problem:Observe(see/watch/look/Exp.)

what happens to a process, what outcomes obtained

when you change certain input factors,

I/O relationships: cause-and-effect or

Understanding->Learning->Study

considerations,choices,decisions,conclusions, attempt/try,effort, guess, adopt, reach, demonstrate, Express,Consider/Consideration,interpret

regard to/as /regardless of, can be regarded as,

describe, show, indicate, illustrate,specify/specific/specification, occur/occurrence

a general process

a particular process or system.

all fields of inquiry, inquire/

almost, virtually, essentially,

related/relate to/relational/relationship/relative: relative frequency,

Should/Must/Would/Could/Shall/Will/May/Might

the true meaning of

according to,

is involved in

serve as

the basis for, is based on,

pertaining to

the foundations of

interpretations of

Suppose, be assumed that

adopt: unless a person is simply willing to adopt a collection of judgements known to be consistent.

reach: two or more scientists working together to reach a common evaluation of the state of knowledge in some scientificarea of common interest.

Experiment: IDENTITY all possible OUTCOMES

Test: Experimental Run

Conclusions(valid and objective)<-Study:

Design

Experiment:

Project: planning

Plan: conducting

resulting data: analyzing

Investigator/Investigate, error/exception/warning, discover

Observe=>Learning=>Study;

1.1 Strategy of Experimentation

Observing a system or process while it is in operation is an important part of the learning process, and is an integral part of understanding and learning about how systems and processes work

The great New York Yankees catcher Yogi Berra said that “. . . you can observe a lot just by watching.”

However, to understand what happens to a process when you change certain input factors, you have to do more than just watch, you actually have to change the factors. This means that to really understand cause-and-effect relationships in a system, you must deliberately change the input variables to the system and observe the changes in the system output that these changes to the inputs produce. In other words, you need to conduct experiments on the system.

Observations on a system or process can lead to theories or hypotheses about what makes the system work, but experiments of the type described above are required to demonstrate that these theories are correct.

Investigators perform experiments in virtually all fields of inquiry, usually to discover something about a particular process or system.

Each experimental run is a test.

More formally,we can define an experiment as a test or series of runs in which purposeful changes are made to the input variables of a process or system so that we may observe and identify the reasons for changes that may be observed in the output response. We may want to determine which input variables are responsible for the observed changes in the response, develop a model relating the response to the important input variables and to use this model for process or system improvement or other decision-making.

This book is about planning and conducting experiments and about analyzing the resulting data so that valid and objective conclusions are obtained. Our focus is on experiments in engineering and science.

Experimentation plays an important role in technology commercialization and product realization activities,

which consist of:

new product design and formulation, manufacturing process development,

and process improvement.

The objective in many cases may be to develop a robust process, that is, a process affected minimally by external sources of variability.

There are also many applications of designed experiments in a nonmanufacturing or non-product-development setting, such as marketing, service operations, and general business operations.

As an example of an experiment, suppose that a metallurgical engineer is interested in studying the effect of two different hardening processes, oil quenching and salt water quenching, on an aluminum alloy.

Here the objective of the experimenter (the engineer) is to determine which quenching solution produces the maximum hardness for this particular alloy.

The engineer decides to subject a number of alloy specimens or test coupons to each quenching medium and measure the hardness of the specimens after quenching.

The average hardness of the specimens treated in each quenching solution will be used to determine which solution is best.

As we consider this simple experiment, a number of important questions come to mind:

1. Are these two solutions the only quenching media of potential interest?
2. Are there any other factors that might affect hardness that should be investigated or controlled in this experiment (such as, the temperature of the quenching media)?
3. How many coupons of alloy should be tested in each quenching solution?
4. How should the test coupons be assigned to the quenching solutions, and in what order should the data be collected?
5. What method of data analysis should be used?
6. What difference in average observed hardness between the two quenching media will be considered important?

All of these questions, and perhaps many others, will have to be answered satisfactorily before the experiment is performed.

Experimentation is a vital part of the scientific (or engineering) method.

Now there are certainly situations where the scientific phenomena are so well understood that useful results including mathematical models can be developed directly by applying these well-understood principles.

The models of such phenomena that follow directly from the physical mechanism are usually called mechanistic models.

A simple example is the familiar equation for current flow in an electrical circuit, Ohm's law, E=IR.

However, most problems in science and engineering require observation of the system at work and experimentation to elucidate information about why and how it works.

Well-designed experiments can often lead to a model of system performance;

such experimentally determined models are called empirical models.

Throughout this book, we will present techniques for turning the results of a designed experiment into an empirical model of the system under study.

These empirical models can be manipulated by a scientist or an engineer just as a mechanistic model can.

A well-designed experiment is important because the results and conclusions that can be drawn from the experiment depend to a large extent on the manner in which the data were collected.

To illustrate this point, suppose that the metallurgical engineer in the above experiment used specimens from one heat in the oil quench and specimens from a second heat in the saltwater quench. Now, when the mean hardness is compared, the engineer is unable to say how much of the observed difference is the result of the quenching media and how much is the result of inherent differences between the heats.

1 Thus, the method of data collection has adversely affected the conclusions that can be drawn from the experiment.

1 A specialist in experimental design would say that the effect of quenching media and heat were confounded; that is, the effects of these two factors cannot be separated.

Grade/Year/Semester / Class

Education plan, in terms of time and plan,

• Grade:
  • Graduate: freshmen ⇒ sophomore ⇒ junior ⇒senior
  • Graduated: Grade 1 master ⇒ Grade 2 master ⇒ Grade 3 master
• Semester:
  
  Year / Fiscal Year / Season

Belief / Judgement

"mind"

mind / remind

they MUST Express the strength of their belief in numerical terms.

Thought

Information

"language"

proposition

statement

argument

proof

interpretation

this section describe THREE COMMON OPERATIONAL interpretation of probability.

"Term"

Difference between "Terms" and "Manner";

Within the "Proba. and Statis." Subject, especially in the "Probability Theory", we often use the following:

in terms of "Set"/"Set Theory",

in terms of "Events",

Basis / Foundation

basis

Foundation:

"Axiom"

Self-Evident and set as the basis/foundation for a Whole Theory;

"Definition"

"Concept"/"ideal"/"Context"/"notation"/"Symbol"/"Term"/"Axiom"/"Definition"/"Condition&

"Condition"

"Property"

"Law"