Math 647Math 647

March 18 & 20, 2002March 18 & 20, 2002

ProbabilityProbability

Probability:Probability: Probability of exceedance: Pr[ X > x ] or Pr Probability of exceedance: Pr[ X > x ] or Pr

[X < x ][X < x ] Relative frequency/proportions: (e.g. 1 out Relative frequency/proportions: (e.g. 1 out

of 4; x twice as likely as y)of 4; x twice as likely as y) Degree of belief: includes huge human bias Degree of belief: includes huge human bias

& error& error Importance of ProbabilityImportance of Probability

Allows us to describe uncertainty in Allows us to describe uncertainty in judgments, outcomes, eventsjudgments, outcomes, events

Issues with ProbabilityIssues with Probability

Forms of Pr:Forms of Pr: LikelihoodLikelihood Error to a quasi-deterministic valueError to a quasi-deterministic value Confidence interval to a valueConfidence interval to a value Range to a variableRange to a variable Probability distributions: models Probability distributions: models

Historical data miningHistorical data mining Forecasts, projectsForecasts, projects

Developing Probabilities Developing Probabilities Requires DataRequires Data

How to gather data?How to gather data? Consider how the random variable should Consider how the random variable should

behave:behave: What are the natural ranges (+/-, bounded by What are the natural ranges (+/-, bounded by

zero and/ or a max/min)zero and/ or a max/min) What is typical for such variables?What is typical for such variables? What does history tell us?What does history tell us?

This requires fitting modelsThis requires fitting models What is the availability of the data?What is the availability of the data? Expert bias: Who knows? What is their Expert bias: Who knows? What is their

motivation? Why would they under/over motivation? Why would they under/over estimate? Ask yourself, would you do the same?estimate? Ask yourself, would you do the same?

Developing the Pr to Useful Developing the Pr to Useful FormsForms

Motivation: focus the statistic. Pr of Motivation: focus the statistic. Pr of what? Be specific? A&A suggests an what? Be specific? A&A suggests an educational phase. When asking for a Pr, educational phase. When asking for a Pr, be sure the expert understand what it be sure the expert understand what it means? means?

Classic example: You either win the Classic example: You either win the lottery or you don’t, so your chances are lottery or you don’t, so your chances are 50-50…right and you are an expert !?! 50-50…right and you are an expert !?!

Structure of the PrStructure of the Pr

Again be specific!Again be specific! State and explain assumptions.State and explain assumptions. Ask why the Pr is suggested.Ask why the Pr is suggested. Are there governing circumstances Are there governing circumstances

that explain the Pr?that explain the Pr?

What’s the probability that next What’s the probability that next summer is as hot or hotter than last? summer is as hot or hotter than last? 100%-I believe in global warming; it 100%-I believe in global warming; it will always get hotter! will always get hotter!

Conditions of PrConditions of Pr

Force the subject to think beyond Force the subject to think beyond his/her experiences.his/her experiences.

This can be challenging.This can be challenging. Ask the subject why they have Ask the subject why they have

selected a range. Ask them by the Pr selected a range. Ask them by the Pr is not larger/smaller.is not larger/smaller.

Why is there a 70% chance that it will Why is there a 70% chance that it will rain tomorrow? Because it hasn’t rained rain tomorrow? Because it hasn’t rained yet, and chance keeps getting higher?yet, and chance keeps getting higher?

Discrete vs. ContinuousDiscrete vs. Continuous

Recall that even continuous variables Recall that even continuous variables are only observed in discrete sample are only observed in discrete sample and intervals. In the limit, many and intervals. In the limit, many continuous variables can be treated as continuous variables can be treated as discrete.discrete.

Some things don’t add value: Expected Some things don’t add value: Expected number of wins next year for the soccer number of wins next year for the soccer team = 15.2 games. Games won remain team = 15.2 games. Games won remain discrete. Consider how the variable in discrete. Consider how the variable in question should be expressed: > 15 question should be expressed: > 15 wins, from 14-17 wins, etc.wins, from 14-17 wins, etc.

Review Valuable Summary Review Valuable Summary StatisticsStatistics

Central tendency:Central tendency: Mean: expectationMean: expectation Median: value with 50% Pr of exceedanceMedian: value with 50% Pr of exceedance Mode: most frequent value. More Mode: most frequent value. More

meaningful in discrete datameaningful in discrete data Percentiles: 1,5,10,25,50,75,90,95,99, Percentiles: 1,5,10,25,50,75,90,95,99,

etc. Ask which end of the distribution is etc. Ask which end of the distribution is critical? Maybe both are critical.critical? Maybe both are critical.

Quartiles: 25Quartiles: 25thth & 75 & 75thth percentiles percentiles

Probability DistributionsProbability Distributions

CDF: cumulative distribution CDF: cumulative distribution functionfunction Must be monotonic & bounded at 1.0 Must be monotonic & bounded at 1.0

and 0.0 on the dependent axis.and 0.0 on the dependent axis. Derived by integrating the PDF Derived by integrating the PDF

(probability density function)(probability density function)

PDFs and PFMsPDFs and PFMs PDF: only for PDF: only for

continuous continuous random variablesrandom variables

PFM: only for PFM: only for discrete variablesdiscrete variables

Such pdfs and Such pdfs and pfms can be pfms can be derived from derived from Monte Carlo Monte Carlo simulations simulations where where appropriate.appropriate.

0 2 4 6 80

0.2

0.4

0.6

0.8

0 10 20 300

0.1

0.2

0 5 10 150

0.05

0.1

0.15

0.2

0.25

PDFs and PMFsPDFs and PMFs

PDF/PMF: describes the distribution of PDF/PMF: describes the distribution of the random variable. Form may be the random variable. Form may be empirical (as in a histogram) or taken empirical (as in a histogram) or taken from a model (normal, gamma, beta, from a model (normal, gamma, beta, etc.)etc.)

Appeal to appropriate models when Appeal to appropriate models when convenient. Height of people follows a convenient. Height of people follows a skewed distribution. Average height for skewed distribution. Average height for multiple groups follows a normal multiple groups follows a normal distribution via central limit theorem.distribution via central limit theorem.

Use of Summary Use of Summary StatisticsStatistics

Verify statistical behavior with easily Verify statistical behavior with easily understandable statistics:understandable statistics: Mean/mode/medianMean/mode/median PercentilesPercentiles Range (Max-Min)Range (Max-Min) Variance/std. dev.Variance/std. dev.

Use of percentiles helps identify and Use of percentiles helps identify and verify statistical forms: symmetry, verify statistical forms: symmetry, skew, bounds, bi-modal, tails, central skew, bounds, bi-modal, tails, central tendencytendency