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```MBA elective - Models for Strategic Planning - Session 14
Choosing Probability Distributions in Simulation
Probability Distributions may be selected on the basis of
Data
Theory
Judgment
a mix of the above
General approach
1. Gather all known information about the uncertain variable
2. Match the information with a probability distribution
© 2009 Ph. Delquié
1
Gathering information about the uncertain variable
Any relevant data available?
If yes, consider using the “Fit...” option of Crystal Ball
Is the variable discrete or continuous?
Want discrete values out of a continuous distribution?
Just round the values to integer =ROUND(Assumption Cell, 0)
Do we know anything about the process underlying the variable?
2
The Uniform Distribution
Almost Complete Ignorance
We know nothing except for a range (Min and Max)
No reason to think that some values are more likely than others
No reason to think that some values are less likely than others
Probability density
Uniform Distribution (Min=40, Max=58)
40.00
44.50
49.00
53.50
58.00
Assessing the range of a variable:
Think of “highest” lower bound, “lowest” upper bound
Beware: Overconfidence range too narrow
3
The Triangular Distribution
We know a “best guess” value
We know the most likely value (the Mode), the Min and the Max
Min, Mode, Max are often based on judgment
We believe: the larger a deviation from central value, the less likely
May have a non symmetric shape
Beware: most likely value ≠ expected value
COST PER UNIT (EUR)
Mean = 11.33
8.00
10.00
12.00
14.00
16.00
Assessing the range of the variable:
Think of bounds separately, not just as deviations from typical value
Beware: Anchoring on Mode to judge Min and Max values range too narrow
4
The Normal Distribution
We believe the variable results from the combination of many
independent random factors, or a natural process
Deviations up or down from the mean are equally likely
We need to specify the Mean and Standard Deviation
Are any percentiles known?
A distribution in Crystal Ball may be defined using percentiles,
e.g. 10th and 90th.
Normal (Mean = 6, Stdev =12)
Stock return with
6% mean and
12% standard
deviation
-30.00
-12.00
6.00
24.00
42.00
Judgmentally, percentiles may be easier to assess than the standard deviation
5
The Binomial Distribution
We know something about the process
We can think of the variable as a number of instances out of a
specified number of trials
E.g.: Any web page viewer has a 10% chance of making a purchase.
We’ll have 100 page viewers. How many sales could we have?
We know the probability of occurrence on any trial (P)
e.g. probability that any particular cell phone will ring during class
We know the total number of trials (N)
e.g. total number of phones that are on ring mode during class
BINOMIAL DISTRIBUTION (P = 0.05, N = 10)
BINOMIAL DISTRIBUTION (P = 0.1, N = 100)
.132
.599
.099
.449
.066
.299
.033
.150
.000
.000
0
6
12
18
24
0
1
2
3
4
6
The Poisson Distribution
We know something about the process
We can think of the variable as a number of instances over a specified time
period, occurring randomly at a given rate
E.g.: Number of help requests arriving at the 4618 computer hotline per hour
We know the average rate of occurrence (e.g. 5 per hour)
Average rate of occurrence is constant
POISSON DISTRIBUTION (RATE = 5)
.175
.132
.088
.044
.000
0.00
3.25
6.50
9.75
13. 00
7
Other Distributions
Other distributions may be used
Theoretical reasons
Lognormal Distribution: the log of the random variable follows a Normal
e.g. stock prices
Beta Distribution: the random variable has natural bounds
e.g. a proportion, probability, market share, etc.
Empirical reasons: it has proved to fit well to past data
Avoid using a distribution that you do not understand
Single event
The =RAND() function returns a value uniformly distributed between 0
and 1. Can be used as part of other Excel formulas
Ex: =(RAND()<0.35) will return 1 with prob. 0.35, 0 with prob. 0.65
8
Mixture Distributions
Modeling “fat” tails with a mixture of normal distributions
Ex:
Use:
Mix a Normal with Mean=0, Stdev=1
with a Normal with Mean=0, Stdev=5
=IF(RAND()>0.5, CB.Normal(0,1), CB.Normal(0,5))
9
Picking the right distribution = Art + Science
All you know
about the
random variable’s
Process,
Range,
Likely values,
Percentiles,
etc&
Features and
properties
match
of common
probability
distributions
. . . or make up a “Custom” distribution
10
Some useful short-cut functions provided by Crystal Ball
Distribution
Excel-style function (active when CB is loaded)
Uniform
=CB.Uniform(Min, Max)
Normal
=CB.Normal(Mean, StdDev)
Triangular
=CB.Triangular(Min, Mode, Max)
Binomial
=CB.Binomial(Prob, Trials)
Poisson
=CB.Poisson(Rate)
Exponential
=CB.Exponential(Rate)
Lognormal
=CB.Lognormal(Mean, StdDev)
Custom
=CB.Custom(Array of values/prob.)
11
Declaring random variables with “CB.functions”
Pros
• quick, easy to use
• provide great flexibility (e.g. can be used in ‘IF’ statements)
• can be combined with any other Excel function
• parameters can be dynamic
Cons
• will not appear as assumptions in the simulation Report
• will not be selected when you ‘Select Assumptions’ in your model
• will not be included in the Sensitivity estimations
• do not allow you to define explicit correlation with other variables
12
For next session
Session 15
Prepare SCOR-eSTORE.COM case
The simulation model is already built
Exploit the model to estimate the value of real options
Individual case (will be accepted until Session 16)
Turn in a hard copy in class as per instructions
13
```