Introduction to Probability and Statistics

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IntroductiontoProbability and Statistics

• Probability and Statistics for Scientists and
  Engineers

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Objective

• Introduction to fundamental probabilistic and
  statistical concepts, methods and techniques
• for application to engineering and science for
• modeling and analysis of variability
• analysis of data
• to impact the decision making process.

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Probability and Statistics

• Objectives
• Model population to mathematically describe
  physical relationships
• Describe variability
• Quantify uncertainty
• Analyze data to increase knowledge draw
  conclusions and make decisions
• Describe data numerically and graphically
• Make inference about underlying population

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Relationship Between Probability Statistics
Probability
Sample
Population
Descriptive Statistics
Inferential Statistics
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Probability

• What is it?
• The term probability refers to the study of
  randomness
• and uncertainty.
• How can it be applied to engineering?
• The Theory of Probability provides methods for
  quantifying the chances, or likelihood,
  associated with the various outcomes in any
  situation in which one of a number of possible
  outcomes may occur.

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Basic Questions

• First, there is a question of what we mean when
  we say that a probability is 0.82, or 0.25.
• What is probability?
• Then, there is the question of how to obtain
  numerical values of probabilities, i.e., how do
  we determine that a certain probability is 0.82,
  or 0.25.
• How is probability determined?
• Finally, there is the question of how
  probabilities can be combined to obtain other
  probabilities.
• What are the rules of probability?

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Purpose of Analysis

• To impact the decision-making process

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The Analysis Process

• Define the problem and formulate an objective
• Identify the analysis options for accomplishing
  the objective and down-select to the preferred
  option
• Perform the analysis and draw conclusions
• Present the analysis results, both technically
  and from the layperson perspective

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Problem Solving Approach

• Define the problem
• Visualize
• Develop diagrams, sketches, tables, etc
• Review the problem definition objective
• Formulate possible alternates for solving the
  problem
• Down select to the preferred method
• Solve the problem
• Review the results and prepare report
• Present results

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Analysis Process
Define Problem and Requirements
Visualize Relate to Physical
Develop Diagram /Flow chart
Establish Ground Rules and Assumptions
Revisit Problem and Define Objective
Identify Solution Approaches
Down-select to preferred Solution Approach
Review Results and Draw Conclusions
Solve Problem and Perform Check
Present Results
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We remember

• 10 of what we read
• 20 of what we hear
• 30 of what we see
• 50 of what we hear see
• 70 of what we say
• 90 of what we say do

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STEP 1 OF THE SCIENTIFIC METHODDEFINING THE
PROBLEM

• by
• Roy E. Rice, Ph.D., P.E.
• AF OR FAM Course
• 25 Apr 2005

Selected Charts
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DEFINING THE PROBLEM
Outline

• Introduction
• The Scientific Method
• The Role of the People involved in the Analysis
• Explanation of established procedures
• The Template
• Examples

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INTRODUCTION
Role of Analysis
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INTRODUCTION
Objective/Question
Data/ Assumptions
Model(s)
Analyst
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THE SCIENTIFIC METHOD
Operations research is a scientific method of
providing executive departments with a
quantitative basis for decisions regarding the
operations under their control. -- Morse and
Kimball

• Define the Problem
• Develop Alternatives
• Develop Criteria to distinguish among
  alternatives
• Develop measures of the criteria
• Determine Analytic Methodology (to include DOE)
• Determine tool(s) to execute the methodology
• Gather data
• Exercise tool(s)
• Analyze results
• Report Results

Typically we jump to these steps
Typically we spend the least amount of our time
in this step
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DEFINE THE PROBLEM

• Step 1
• Define Problem
• Step 1a (Roys Rule)
• Define the Problem againbecause you didnt do
  it right the first time!
• Step 1b (Vince Roskes Rule)
• First find out what the question is - then find
  out what the real question is!
• EXAMPLE Close Air Support Study in J8

It often occurs that the major contribution of
the operations research worker is to decide what
is the real problem. -- Morse and Kimball
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DEFINE THE PROBLEM- WHY? -

• Perfectly right answer to the totally wrong
  question!
• Probably waste resources
• Probably lose Credibility
• NATO Code Of Best Prictices - DRAFT
• In the initial problem formulation iteration, it
  is critical to begin with an understanding of the
  REAL problem rather than a determination to apply
  readily available tools, scenarios, and data.
• Explicit problem formulation must precede
  construction of concepts for analysis or method
  selectionProper resourcing of problem
  formulation activities will improve the overall
  efficiency and quality of the study.
• An understanding of the decisions to be
  supported by the analysis and the viewpoints of
  the various stakeholders is essential to
  clarifying the study issues.

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THE ROLE OF THE PEOPLE INVOLVED IN PROBLEM
DEFINITION

• The Decision Maker
• Communicate the context of the Problem
• Buy into the Problem and Solution Approach
• Provide Heading Checks
• Have Open Door
• Commitment vs. Participation - ham eggs
• The Mid-level Manager
• Be a Conduit
• What does your FINAL chart look like?
• Must understand analysis - what it can/cannot do
• The Analyst
• Articulate the Problem back to D.M.
• Must put himself/herself in the D.M.s shoes

Problem Definition is an Iterative Process
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EXPLANATION OF ESTABLISHED PROCEDURES
Principles of Management Science - Wagner

• Initial Diagnosis
• Identify critical factors
• What the principle decisions are
• What the MOEs( Measures of Effectiveness) are
• Tradeoffs to compare alternatives
• Problem Elements
• Controllable or decision variables
• Uncontrollable variables
• Restrictions or constraints on the variables
• Objectives for defining a good or improved
  solution
• Confines of the analysis - multifold impacts

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EXPLANATION OF ESTABLISHED PROCEDURES
Naval War College Pub

• Decision Objectives - goal of the decision
• Problem Context
• Stakeholders
• Triggers
• Influences
• Problem Boundaries
• Timeframe
• Rule sets
• Facts
• Assumptions
• Analytic Objectives

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EXPLANATION OF ESTABLISHED PROCEDURES

• Rices Method (from Dr. Bill Lesso, University of
  Texas at Austin)

Given The problem is to By choice
of Subject to
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RICES METHOD
Given set the environment state your
assumptions The problem is to clear statement
of the specific problem to be solved By choice
of determine the decision variables Subject
to determine the constraints and restrictions
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RICES METHOD
Given - what is the timeframe of the study? -
what scenario(s) are applicable? - what
alternatives are to be used? - what are fixed
parameters? - who are the players? - what are the
threats?
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RICES METHOD
Given The problem is to - not to analyze
or to study - not a statement of fact (e.g.,
we have a pilot shortage) - address the
causes not the symptoms - what are you really
trying to quantify or measure? - probably
related to operational objectives - what...
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RICES METHOD
Given The problem is to By choice of -
what are the decision variables? - what does the
decision maker get to play with? - what does
he/she have control over? - what...
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RICES METHOD
Given The problem is to By choice of Subject
to - constraints - hard inviolable (e.g.,
distances, 24 hours in a day, ramp space) -
restrictions - man-made (e.g., start times) -
what are the constraints on the variables? - what
might you want to perform sensitivity analysis
on?
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SUMMARY

• Follow the Scientific Method
• Step 1 Problem Definition is the most critical
  step
• Problem Definition is an iterative process
• Must get Decision Maker commitment
• Procedures
• NATO C2 COBP
• Rices Method

In section 3.4.4 is reported a case where it was
nearly decided that it was not worth while to put
antiaircraft guns on merchant vessels because
they did not shoot down enemy planes. It took an
operations research worker to point out that,
even though the enemy planes were not shot down,
the antiaircraft guns were valuable because they
decreased the accuracy of the enemy planes enough
to lessen the chance that the merchant vessel be
sunk.. -- Morse and Kimball