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Random Number Generation

Dr. Jerrell T. Stracener, SAE FellowUpdate: 1/31/02

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Generating Random Numbers

Generating values of a random variable using theprobability integral transformation to generate arandom value y from a given probability densityfunction f(y):

1. Generate a random value rU from a uniformdistribution over (0, 1).

2. Set rU = F(y)

3. Solve the resulting expression for y.

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Generating Random Numbers with Excel

From the Tools menu, look for Data Analysis.

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Generating Random Numbers with Excel

If it is not there, you must install it.

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Generating Random Numbers with Excel

Once you select Data Analysis, the following window will appear. Scroll down to “Random Number Generation” and select it, then press “OK”

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Generating Random Numbers with Excel

We would like U(0, 1). So select “Uniform” under the “Distribution” menu.Type in “1” for number of variables and 10 for number of random numbers. Then press OK. 10 random numbers of uniform distribution will now appear on a new chart.

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Generating Random Numbers

f(y)

F(y)y

y

1 2

3

1.00.80.60.40.2 0

ri

yi

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Generating Random Values from the ExponentialDistribution E()

• generate ri from U(0, 1)

• calculatexi = - ln(1 - ri)

• sincefor i = 1, 2, …, n

ix

i er

1

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Generating an Exponential Distribution with Excel

See charts 7 - 10.

Select a that you would like to use, we will use = 5.

Type in the equation -ln(1 - ri), with filling in as 5, and ri as cell A1. Now with that cell selected, place the cursor over the bottom right hand corner of the cell. A cross will appear, drag this cross down to B10. This will transfer that equation to the cells below. Now we have an exponential distribution in cells B1 - B10.

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Generating Random Values from the WeibullDistribution W(, )

• generate ri from U(0, 1)

• calculatexi = [-ln(1 - ri)]1/

• since for i = 1, 2, …, n

ix

i er 1

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Generating a Weibull Distribution with Excel

See charts 7 - 10.

Select a and that you would like to use, we will use = 100, = 20.

Type in the equation xi = [-ln(1 - ri)]1/, with filling in as 100, as 20, and ri as cell A1. Now transfer that equation to the cells below. Now we have an Weibull distribution in cells B1 - B10.

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Generating Random Values from the LognormalDistribution Ln(, )

• generate ri from N(0, 1)

• calculate

• sinceLn xi = + ri for i = 1, 2, …, n

iri ex

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Generating a Lognormal Distribution with Excel

See charts 7 - 10.

Select a and that you would like to use, we will use = 2, = 1.

Type in the equation , with filling in as 2, as 1, and ri as cell A1. Now transfer that equation to the cells below. Now we have an Lognormal distribution in cells B1 - B10.

iri ex