Copyright © 2010 Lumina Decision Systems, Inc. Common Parametric Distributions Gentle Introduction...

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 Webinar10 June 2010


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


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…


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


Distribution Types

• Discrete

• Continuous


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)


ChanceDist Exercise

An event occurs on one of the 7 days of the week.

• Each weekday 8%• Each day of weekend 30%

Create a chance variable named Day_of_event with this distribution.


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.


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.


Custom (non-parametric)Continuous Distributions

• CumDist(p,x,i) Parameters:

p : Probabilities that value <= xx : Ascending set of valuesi : index shared

CumDist(p,x,x) or just CumDist(p,x)


CumDist Exercise

• A geologist estimates the capacity of a recently discovered oil deposit. He expresses is assessments as follows:

100% that 100K < capacity < 1B barrels90% that 5M < capacity < 500M barrels75% that 50M < capacity < 100M barrelsMedian estimate: 75M barrels

• Use CumDist to encode these estimates as a distribution for capacity.


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..143Variable 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.


The Most Commonly used Parametric Distributions

• Discrete:BernoulliPoissonBinomialUniform integer

• Continuous:NormalLogNormalUniformTriangularExponentialGammaBeta


Why chose one distribution over another?

• Discrete or continuous?• Bounded quantity or infinite tails?

Bounded both sides

One-sidedtail

Two tailed

Continuous

UniformTriangularBeta

LogNormalGammaExponential

NormalStudentTLogistic

Discrete BinomialUniform int

Poisson


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 propertiesfor mathematicians – not model builders.

• CorrectnessOther than broad properties, the sensitivity of computed results to specific choice of distributions for assessments is usually extremely low.

x


Distributions forInteger-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.


Distributions forInteger-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.


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.


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.


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?


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.


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.


Bounded comparisons

• Using:Min = 10Mode = 25Max = 40

• Compare distributions (on same PDF & CDF plot):

TriangularPertUniform

• Repeat for Mode=15


Central Limit Theorem

• Suppose y = x1·x2·x3· .. ·xN

z = x1+x2+x3+ .. +xN

Each xi ~ P(·), where P(·) is any distribution. (each xi is independent)

• Then as N→∞, y→LogNormal(..)z→Normal(..)


Sensitivity to Distribution Choice

• Load the TXC model (Example Models – Risk Analysis)

• Compare Total_cost for these Control_cost_factor distributions:

LogNormal(mean:108.6M,stddev:45.96M)Gamma(5.58,19.45M)Uniform(29M,188M)Triangular(41M,60M,245M)Weibull(2.53,122.4M)

• Using the LogNormal:Compare Total_cost when Control_cost_factor mean is increased or decreased by 10%.Compare when stddev is altered by 50%


Summary

• Various parametric distributions are convenient for certain type of quantities.

• Choice of parametric distribution is usually driven by:

Continuous vs. discreteTails or boundedBroad shapeType of information easily estimated

• Results are usually fairly insensitive to exact choice of distribution type.

Copyright © 2010 Lumina Decision Systems, Inc. Common Parametric Distributions Gentle Introduction...

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