1. About the Exam No cheat sheet Bring a calculator.
You may NOT use the calculator on your phone or iPad or computer or other media device
Short essay answers
Math problems to be solved
Know all materials covered including last Thursday’s lecture on simulation
Materials presented in Lab can also be include in the exam

2. Materials for Lecture 14 Chapters 4 and 5
Chapter 16 Sections
Lecture 14 Bernoulli .xlsx
Lecture 14 Normality Test.xlsx
Lecture 14 Simulation Model with Simetar.xlsx
Lecture 14 Normal.xlsx
Lecture 14 Simulate a Reg Model.xlsx

3. Stochastic Simulation
Purpose of simulation is to estimate the unknown probability distribution for a KOV so decision makers can make a better decision
We simulate because we cannot observe and measure the KOV distribution directly
We want to test alternative values for control variables
Business simulation models are basically an equation to calculate profit
Profit = (Price * Quantity) – (VC * Quantity) - FC
Profit = * - VC * FC

4. Stochastic Simulation
Simulation sample PDFs for random variables, calculate values of KOV for many iterations
Record KOV over many iterations
Analyze KOV distribution

5. Stochastic Variables
Any variable the decision maker cannot control is a stochastic variable
In agriculture we think of yield as stochastic as it is subject to weather
For most businesses, the prices of inputs and outputs are not directly controlled by management so they are stochastic.
Production may be random as well.
Include the most important stochastic variables in simulation models
Your model can not include all random variables

6. Stochastic Simulation
In economics we use simulation because we can not experiment on live subjects, a business, or the economy without injury
In other fields they can create an experiment
Health sciences they feed (or treat) lots of lab rats on different chemicals to see the results
Animal science researchers feed multiple pens of steers, chickens, cows, etc. on different rations
Engineers run a motor under different controlled situations (temp, RPMs, lubricants, fuel mixes)
Vets treat different pens of animals with different meds
Agronomists set up randomized block treatments for a particular seed variety with different fertilizer levels
All of these are just different iterations of “models”

7. Iterations, How Many are Enough?
Specify the output variables’ names and location
Specify the number of iterations in the Simetar simulation engine
Change the number of iterations based on the nature of the problem is adequate.
Some studies use 1,000’s because they are using a Monte Carlo sampling procedure which is less precise than the Latin hypercube
Simetar uses a Latin hypercube so 500 is an adequate sample size

8. Definitions
Stochastic Model – means the model has at least one random variable
Monte Carlo simulation model – same as a stochastic simulation model
Two ways to sample or simulate random values
Monte Carlo – draw random values for the variables purely at random
Latin Hyper Cube – draw random values using a systematic approach so we are certain that we sample ALL regions of the probability distribution
Monte Carlo sampling requires larger number of iterations to insure that model samples all regions of the probability distribution
For a U(0,1) the CDF is straight line
MC has bias from straight line
LHC is the straight line
This is with 500 iterations
Simetar default is LHC

9. Simulating Random Variables
Normal distribution used frequently, particularly when simulating residuals for a regression model
Parameters for a Normal distribution
Mean expressed as Ῡ or Ŷ
Standard Deviation σ
Assume yield is a random variable and we have a production function, such as:
Ỹ = a + b1 Fertilizer + b2 Water + ẽ
Deterministic component is: a + b1 Fertilizer + b2 Water
Stochastic component is: ẽ
Stochastic component, ẽ, is assumed to be distributed Normal
Mean of zero
Standard deviation of σe
See Lecture 14 Simulate a Reg Model.XLSX

10. Use the Normal Distribution When:
Use the Normal distribution if you have lots of observations and have tested for normality
Watch for infeasible values from a Normal distribution (negative yields and prices)

11. Problems with the Normal
It is easy to use, so it often used when it is not appropriate
It does not allow for extreme events (Black Swans)
No way to account for record breaking outliers because the distribution is defined by Mean and Std Dev.
Std Dev is the “average” deviation from the mean and averages out BS’s
Market outliers are washed away in the average
It is the foundation for Sigma 6
So Sigma 6 suffers from all of the problems above
Creates a false sense of security because it never sees a record braking outlier

12. How to Test for Normality
Simetar provides an easy to use procedure for testing Normality that includes:
S-W – Shapiro-Wilks
A-D – Anderson-Darling
CvM – Cramer-von Mises
K-S – Kolmogornov-Smiroff
Chi-Squared
Simetar’s Hypothesis Testing Icon provides a tab to “Test for Normality”

13. Simulating a Normal Distribution
=NORM( Mean, Standard Deviation)
=NORM( 10,3)
=NORM( A1, A2)
Standard Normal Deviate (SND)
=NORM(0,1) or =NORM()
SND is the Z-score for a standard normal distribution allowing you to simulate any Normal distribution
SND is used as follows:
Ỹ = Mean + Standard Deviation * NORM(0,1)
Ỹ = Mean + Standard Deviation * SND
Ỹ = A1 + (A2 * A3) where a SND is in cell A3

14. Truncated Normal Distribution
General formula for the Truncated Normal
=TNORM( Mean, Std Dev, [Min], [Max],[USD] )
Truncated Downside only
=TNORM( 10, 3, 5)
Truncated Upside only
=TNORM( 10, 3, , 15)
Truncated Both ends
=TNORM( 10, 3, 5, 15)
Truncated both ends with a USD in general form
=TNORM( 10, 3, 5, 15, [USD])
The values in the [ ] are optional

15. Example Model of Net Returns for a Business Model
- Stochastic Variables -- Yield and Price
- Management Variables -- Acreage and Costs (fixed and variable)
- KOV -- Net Returns
- Write out the equations and exogenous values
Equations and their order

16. Program a Simulation Model in Excel/Simetar -- Input Data Section of the Worksheet
See Lecture 14 Simulation Model with Simetar.XLSX

17. Program Model in Excel/Simetar -- Generate Random Variables and Simulate Profit

18. Bernoulli Distribution1 X
PDF for Bernoulli B(0.75)
.25
.75
CDF for Bernoulli B(0.75)
PDF and CDF for a Bernoulli Distribution.
Parameter is ‘p’ or the probability that the random variable is 1 or TRUE
Simulate Bernoulli in Simetar as
= Bernoulli(p)
= Bernoulli(0.25)
Lecture 14 Bernoulli.XLSX examples follow

19. Bernoulli Distribution Application

20. Bernoulli Distribution Application