1. Math 647 March 18 & 20, 2002

2. Probability Probability: Probability: Probability of exceedance: Pr[ X > x ] or Pr [X x ] or Pr [X < x ] Relative frequency/proportions: (e.g. 1 out of 4; x twice as likely as y) Relative frequency/proportions: (e.g. 1 out of 4; x twice as likely as y) Degree of belief: includes huge human bias & error Degree of belief: includes huge human bias & error Importance of Probability Importance of Probability Allows us to describe uncertainty in judgments, outcomes, events Allows us to describe uncertainty in judgments, outcomes, events

3. Issues with Probability Forms of Pr: Forms of Pr: Likelihood Likelihood Error to a quasi-deterministic value Error to a quasi-deterministic value Confidence interval to a value Confidence interval to a value Range to a variable Range to a variable Probability distributions: models Probability distributions: models Historical data mining Historical data mining Forecasts, projects Forecasts, projects

4. Developing Probabilities Requires Data How to gather data? How to gather data? Consider how the random variable should behave: Consider how the random variable should behave: What are the natural ranges (+/-, bounded by zero and/ or a max/min) What are the natural ranges (+/-, bounded by 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 models This requires fitting models What is the availability of the data? What is the availability of the data? Expert bias: Who knows? What is their motivation? Why would they under/over estimate? Ask yourself, would you do the same? Expert bias: Who knows? What is their motivation? Why would they under/over estimate? Ask yourself, would you do the same?

5. Developing the Pr to Useful Forms Motivation: focus the statistic. Pr of what? Be specific? A&A suggests an educational phase. When asking for a Pr, be sure the expert understand what it means? Motivation: focus the statistic. Pr of what? Be specific? A&A suggests an educational phase. When asking for a Pr, be sure the expert understand what it means? Classic example: You either win the lottery or you don’t, so your chances are 50-50…right and you are an expert !?! Classic example: You either win the lottery or you don’t, so your chances are 50-50…right and you are an expert !?!

6. Structure 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 that explain the Pr? Are there governing circumstances that explain the Pr? What’s the probability that next summer is as hot or hotter than last? 100%-I believe in global warming; it will always get hotter! What’s the probability that next summer is as hot or hotter than last? 100%-I believe in global warming; it will always get hotter!

7. Conditions of Pr Force the subject to think beyond his/her experiences. Force the subject to think beyond his/her experiences. This can be challenging. This can be challenging. Ask the subject why they have selected a range. Ask them by the Pr is not larger/smaller. Ask the subject why they have selected a range. Ask them by the Pr is not larger/smaller. Why is there a 70% chance that it will rain tomorrow? Because it hasn’t rained yet, and chance keeps getting higher?

8. Discrete vs. Continuous Recall that even continuous variables are only observed in discrete sample and intervals. In the limit, many continuous variables can be treated as discrete. Recall that even continuous variables are only observed in discrete sample and intervals. In the limit, many continuous variables can be treated as discrete. Some things don’t add value: Expected number of wins next year for the soccer team = 15.2 games. Games won remain discrete. Consider how the variable in question should be expressed: > 15 wins, from 14-17 wins, etc. Some things don’t add value: Expected number of wins next year for the soccer team = 15.2 games. Games won remain discrete. Consider how the variable in question should be expressed: > 15 wins, from 14-17 wins, etc.

9. Review Valuable Summary Statistics Central tendency: Central tendency: Mean: expectation Mean: expectation Median: value with 50% Pr of exceedance Median: value with 50% Pr of exceedance Mode: most frequent value. More meaningful in discrete data Mode: most frequent value. More meaningful in discrete data Percentiles: 1,5,10,25,50,75,90,95,99, etc. Ask which end of the distribution is critical? Maybe both are critical. Percentiles: 1,5,10,25,50,75,90,95,99, etc. Ask which end of the distribution is critical? Maybe both are critical. Quartiles: 25 th & 75 th percentiles Quartiles: 25 th & 75 th percentiles

10. Probability Distributions CDF: cumulative distribution function CDF: cumulative distribution function Must be monotonic & bounded at 1.0 and 0.0 on the dependent axis. Must be monotonic & bounded at 1.0 and 0.0 on the dependent axis. Derived by integrating the PDF (probability density function) Derived by integrating the PDF (probability density function)

11. PDFs and PFMs PDF: only for continuous random variables PDF: only for continuous random variables PFM: only for discrete variables PFM: only for discrete variables Such pdfs and pfms can be derived from Monte Carlo simulations where appropriate. Such pdfs and pfms can be derived from Monte Carlo simulations where appropriate.

12. PDFs and PMFs PDF/PMF: describes the distribution of the random variable. Form may be empirical (as in a histogram) or taken from a model (normal, gamma, beta, etc.) PDF/PMF: describes the distribution of the random variable. Form may be empirical (as in a histogram) or taken from a model (normal, gamma, beta, etc.) Appeal to appropriate models when convenient. Height of people follows a skewed distribution. Average height for multiple groups follows a normal distribution via central limit theorem. Appeal to appropriate models when convenient. Height of people follows a skewed distribution. Average height for multiple groups follows a normal distribution via central limit theorem.

13. Use of Summary Statistics Verify statistical behavior with easily understandable statistics: Verify statistical behavior with easily understandable statistics: Mean/mode/median Mean/mode/median Percentiles Percentiles Range (Max-Min) Range (Max-Min) Variance/std. dev. Variance/std. dev. Use of percentiles helps identify and verify statistical forms: symmetry, skew, bounds, bi- modal, tails, central tendency Use of percentiles helps identify and verify statistical forms: symmetry, skew, bounds, bi- modal, tails, central tendency