Common Parametric Distributions
Gentle Introduction to Modeling Uncertainty Series #6
Lonnie Chrisman, Ph.D.
Lumina Decision Systems
Analytica Users Group Webinar
10 June 2010
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
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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…
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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
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Distribution Types
• Discrete
• Continuous
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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)
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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.
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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.
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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.
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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)
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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.
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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.
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The Most Commonly used
Parametric Distributions
• Discrete:
Bernoulli
Poisson
Binomial
Uniform
integer
• Continuous:
Normal
LogNormal
Uniform
Triangular
Exponential
Gamma
Beta
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Why chose one distribution
over another?
• Discrete or continuous?
• Bounded quantity or infinite tails?
Bounded
both sides
Continuous Uniform
Triangular
Beta
Discrete
Binomial
Uniform int
One-sided
tail
Two tailed
LogNormal
Gamma
Exponential
Normal
StudentT
Logistic
Poisson
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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.
x
• Correctness
Other than broad properties, the sensitivity of
computed results to specific choice of
distributions for assessments is usually extremely
low.
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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.
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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.
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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.
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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.
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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?
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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.
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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.
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Bounded comparisons
• Using:
Min = 10
Mode = 25
Max = 40
• Compare distributions (on same PDF &
CDF plot):
Triangular
Pert
Uniform
• Repeat for Mode=15
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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(..)
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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%
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Summary
• Various parametric distributions are
convenient for certain type of quantities.
• Choice of parametric distribution is usually
driven by:
Continuous vs. discrete
Tails or bounded
Broad shape
Type of information easily estimated
• Results are usually fairly insensitive to exact
choice of distribution type.
Copyright © 2010 Lumina Decision Systems, Inc.