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Monte Carlo Simulation

Monte Carlo simulation basics

• Random number generator • Start with unit uniform random number (r)

• r is uniformly distributed from 0 to 1

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

X

f(x)

Unit, Uniform random number generator

• Excel • RAND()

• Basic • RND

• Do not use random between • There is not enough discrimination

Procedure for generating 1000 Uniform random numbers in Excel

A1:A1000

Step 1 Step 2 Step 3

Input formula for random number "=Rand()" in cell A1

Copy and paste the formula from A1 and type here A1:A1000, press <ENTER> twice.

1000 Uniform random numbers are generated. Press F9 several times to generate a new set.

Exercise: Simulate flipping a coin

• Simulate • 10 coin flips

• 100 coin flips

• 1000 coin flips

• 10 000 coin flips

• Use the running average

• How many coin flips do you need to be within 1% deviation: 49% and 51%

How many trials?

• Steady-state • A steady-state solution is reached when the output of the simulation from one iteration

to the next changes negligibly

• When calculating means and variances: 1 000 iterations is usually sufficient

• If calculating confidence limits, many more iterations are required; after all, for 99% confidence limits the sample size for the number of random deviates exceeding the confidence limit is 1/100th the number of iterations

• How many coin flips are needed to reach steady-state?

• What is your error?

• Take coin flipping example to steady state

Simulation of 1000 coin flips in Excel

Use formula "=IF(RAND()<0.5,1,0)"

Generating random numbers

• Set r equal to F(x) for the desired distribution and solve for x

• Exponential distribution example

ln(r) 1

-x

r)-ln(1 -x

x-r)-ln(1

er-1

e-1r x

x-

λ

λ

λ

λ

λ

=

=

=

=

= −

Normal random number generators

• Excel • Use the Normal inverse function

• =NORM.INV(RAND(),MEAN,STD)

• Slow with large data sets

• Uses numerical integration

• Fast alternative <Random2.xls>

( ) μσπ += )rcos(2)2ln(r-x 21

Simulate a Normal random variable

• Mean = 50

• Standard deviation = 4

• Generate 1000 random numbers with Excel

• Verify in Jasp or another software package

• How many outliers?

Circuit example

•P = V2/R • Voltage (V) is Normally distributed

• mean = 12

• standard deviation = 0.1

• Resistance (R) is Normally distributed • mean = 2

• standard deviation = 0.2

Where does input data come from?

• Current production

• Supplier

• Discuss how to obtain mean and standard deviation for voltage and resistance

Circuit example

1. What does the power distribution look like? • Specifications

• 72 ± 10

• Will this circuit design meet specifications?

2. Change standard deviation of voltage to 0.3 and standard deviation of resistance to 0.4

• What is the mean and standard deviation of power?

3. Change standard deviation of voltage to 0.5 and standard deviation of resistance to 0.6

Summary

• Simulation can be used to model systems when inputs are known

• Verify all inputs

• Validate models

• Models may be useful even if incomplete