1. Copyright © 2010 Lumina Decision Systems, Inc. Common Parametric Distributions Gentle Introduction to Modeling Uncertainty Series #6 Lonnie Chrisman, Ph.D. Lumina Decision Systems Analytica Users Group Webinar 10 June 2010

2. Copyright © 2010 Lumina Decision Systems, Inc. Course Syllabus Over the coming weeks: What is uncertainty? Probability. Probability Distributions Monte Carlo Sampling Measures of Risk and Utility Risk analysis for portfolios Common parametric distributions Assessment of Uncertainty Hypothesis testing

3. Copyright © 2010 Lumina Decision Systems, Inc. Today’s Topics Continuous vs. discrete. Non-parametric distributions. A handful of the most common distributions. The cases where each is useful. How to encode each in Analytica. Lots of model building exercises…

4. Copyright © 2010 Lumina Decision Systems, Inc. Outline (Order of exercises) “Pre-test” questions Discrete non-parametric: Monte Hall game Continuous non-parametric: Data resampling Event counts: Durations between events Uncertain percentages Bounded Bell shapes

5. Copyright © 2010 Lumina Decision Systems, Inc. Distribution Types Discrete Continuous

6. Copyright © 2010 Lumina Decision Systems, Inc. Custom (Non-parametric) Discrete ChanceDist(P,A,I) Parameters: P = Array of probabilities. Sum(P,I)=1 A = Array of possible outcomes I = Index shared by P and A Note: When A is the index, you can use: ChanceDist(P,A)

7. Copyright © 2010 Lumina Decision Systems, Inc. ChanceDist Exercise An event occurs on one of the 7 days of the week. Each weekday  8% Each day of weekend  20% Create a chance variable named Day_of_event with this distribution.

8. Copyright © 2010 Lumina Decision Systems, Inc. ChanceDist Exercise 2: Monte Hall Game You are a contestant on a game show. A prize is hidden behind 1 of three curtains. You select curtain 1. “Before opening your curtain,” says the host, “let me reveal one of the unselected curtains that does not contain the prize… Curtain 2 is empty! Would you now like to change curtains?” Task: Build an Analytica model, computing the probability of winning the prize if you do or do not change curtains.

9. Copyright © 2010 Lumina Decision Systems, Inc. Monte Hall Steps 1.Chance: Start with the uncertain real location of the prize. 2.Model how the host decides which curtain to show you. He will never reveal the prize or your selected curtain. Otherwise he picks randomly. 3.Decision: Change or not? 4.Objective: Probability that your final selection is the one with the prize.

10. Copyright © 2010 Lumina Decision Systems, Inc. Custom (non-parametric) Continuous Distributions CumDist(p,x,i) Parameters: p : Probabilities that value <= x x : Ascending set of values i : index shared CumDist(p,x,x) or just CumDist(p,x)

11. Copyright © 2010 Lumina Decision Systems, Inc. CumDist Exercise A geologist estimates the capacity of a recently discovered oil deposit. He expresses is assessments as follows: 100% that 100K < capacity < 1B barrels 90% that 5M < capacity < 500M barrels 75% that 50M < capacity < 100M barrels Median estimate: 75M barrels Use CumDist to encode these estimates as a distribution for capacity.

12. Copyright © 2010 Lumina Decision Systems, Inc. Homework challenge: Using CumDist to Resample You have 143 measured values of a quantities. Define an uncertain variable with the same implied distribution (even though your sample size doesn’t match). Here is your synthetic data: Index Data_i := 1..143 Variable Data := ArcCos(Random( over:data_i)) Steps (the parameters to CumDist): Sort Data in ascending order: Sort(Data,Data_i) Compute p – equal probability steps along Data_I, starting at 0 and ending at 1.

13. Copyright © 2010 Lumina Decision Systems, Inc. The Most Commonly used Parametric Distributions Discrete: Bernoulli Poisson Binomial Uniform integer Continuous: Normal LogNormal Uniform Triangular Exponential Gamma Beta

14. Copyright © 2010 Lumina Decision Systems, Inc. Why chose one distribution over another? Discrete or continuous? Bounded quantity or infinite tails? Bounded both sides One-sided tail Two tailed Continuous Uniform Triangular Beta LogNormal Gamma Exponential Normal StudentT Logistic Discrete Binomial Uniform int Poisson

15. Copyright © 2010 Lumina Decision Systems, Inc. Why chose one distribution over another? Discrete or continuous? Bounded quantity or infinite tails? Convenience Some distributions are more “natural” for certain types of quantities. Ease of assessment. Analytical properties for mathematicians – not model builders. Correctness Other than broad properties, the sensitivity of computed results to specific choice of distributions for assessments is usually extremely low. x

16. Copyright © 2010 Lumina Decision Systems, Inc. Distributions for Integer-valued Counts #1 Poisson(mean) Count of events per unit time. # Earthquakes >6.0 in a given year # Vehicles that pass in a given hour # Alarms in a given month # Pelicans rescued from oil spill today When the occurrence of each event is independent of the time of occurrence of other events, the # of occurrences in any given window is Poisson distributed.

17. Copyright © 2010 Lumina Decision Systems, Inc. Distributions for Integer-valued Counts #2 Binomial(n,p) Number of times an event occurs in n repeated independent trials, each having probability p. # oil well blowouts in the next 100 deep-water wells drilled. # people that visit a store in its first month out of the 10,000 residents of the town. # of positive test results in 50 samples tested.

18. Copyright © 2010 Lumina Decision Systems, Inc. Exercise with event counts In a certain region, malaria infections occur at an average rate of 500 infections per year. 10% of infections are fatal. Build an Analytica model to compute the distribution for the number of people expected to die from a malaria infection in a given year.

19. Copyright © 2010 Lumina Decision Systems, Inc. Duration between events Exponential(rate) When events occur independently at a given rate, this gives the time between successive events. Note: rate = 1 / meanArrivalTime Gamma(a,1/rate) Time for a independent events to occur, each having a mean arrival time of 1/rate.

20. Copyright © 2010 Lumina Decision Systems, Inc. Arrival times exercise Cars arrive at a stoplight at a rate of 5 per minute. There is room for 10 cars before nearby freeway traffic is blocked. Graph the CDF for the amount of time until cars begin to block freeway traffic when the light is red. If the light stays red for 90 seconds, what fraction of red light-change cycles will result in blocked traffic?

21. Copyright © 2010 Lumina Decision Systems, Inc. Uncertain Percentages Beta(a,b) Useful for modeling uncertainty about a probability or percentage. Beta(a,b) expresses uncertainty on a [0,1] bounded quantity. Suppose you’ve seen s true instances out of n observations, with no further information. You’d estimate the true proportion as p=s/n. The uncertainty in this estimate can be modeled as: Beta(s+1,n-s+1) Exercise: Of 100 sampled voters, 55 supported Candidate A. Model the uncertainty on the true proportion.

22. Copyright © 2010 Lumina Decision Systems, Inc. Bounded Distributions Triangular(min,mode,max) Often very convenient & natural for expressing estimates when only the range and a best guess are available. Pert(min,mode,max) Same idea as Triangular. To use, include “Distribution Variations.ana” Uniform(min,max) All values between are equally likely. Uniform(min,max,integer:true) All integer values are equally likely.

23. Copyright © 2010 Lumina Decision Systems, Inc. Bounded comparisons Using: Min = 10 Mode = 15 Max = 40 Compare distributions (on same PDF & CDF plot): Triangular Pert Uniform

24. Copyright © 2010 Lumina Decision Systems, Inc. Central Limit Theorem Suppose y = x 1 ·x 2 ·x 3 ·.. ·x N Each x i ~ P(·), where P(·) is any distribution. (each x i is independent) As N → ∞, y’s distribution approaches a LogNormal(..) distribution. Example: Visualize the change in a stock price as the product of zillions of independent disturbances, each disturbance changing it by some percentage. Because of this, “bell curve” shaped distributions are both common and natural.