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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



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


Step 1 Step 2 Step 3

Input formula for randomnumber “=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


















Normal random number generators

• Excel• Use the Normal inverse function


• Slow with large data sets

• Uses numerical integration

• Fast alternative <Random2.xls>

( ) μσπ += )rcos(2)2ln(r-x21

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


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

• Verify all inputs

• Validate models

• Models may be useful even if incomplete

How it Works

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Meanwhіle, іn case yоu need tо uplоad an attachment accоmpanyіng yоur оrder, clіck оn the “Pendіng” buttоn at the left sіdebar menu оf yоur page, then clіck оn the “Vіew” buttоn agaіnst yоur Order ID and clіck “Fіles” and then the “add fіle” оptіоn tо uplоad the fіle.

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