Chapter 8: RandomVariant Generation

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Random-Variate Generation
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Purpose Overview

• Develop understanding of generating samples from
  a specified distribution as input to a simulation
  model.
• Illustrate some widely-used techniques for
  generating random variates.
• Inverse-transform technique
• Convolution technique
• Acceptance-rejection technique
• A special technique for normal distribution

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Inverse-transform Technique

• The concept
• For cdf function r F(x)
• Generate R sample from uniform (0,1)
• Find X sample

X F-1(R)
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Exponential Distribution Inverse-transform

• Exponential Distribution
• Exponential cdf
• To generate X1, X2, X3 , generate R1, R2, R3 ,

r F(x) 1 e-?x for x ??0
Xi F-1(Ri) -(1/?? ln(1-Ri)
Figure Inverse-transform technique for exp(? 1)
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Exponential Distribution Inverse-transform

• Example Generate 200 variates Xi with
  distribution exp(? 1)
• Matlab Code
• for i1200,
• expnum(i)-log(rand(1))
• end

R and (1 R) have U(0,1) distribution
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Uniform Distribution Inverse-transform

• Uniform Distribution
• Uniform cdf
• To generate X1, X2, X3 ,, generate R1, R2, R3 ,

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Uniform Distribution Inverse-transform

• Example Generate 500 variates Xi with
  distribution Uniform (3,8)
• Matlab Code
• for i1500,
• uninum(i)35rand(1)
• end

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Empirical Continuous Distn Inverse-transform

• When theoretical distribution is not applicable
• To collect empirical data
• Resample the observed data
• Interpolate between observed data points to fill
  in the gaps
• For a small sample set (size n)
• Arrange the data from smallest to largest
• Assign the probability 1/n to each interval
• where

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Empirical Continuous Distn Inverse-transform

• Example Suppose the data collected for 100
  broken-widget repair times are

Consider R1 0.83 c3 0.66 lt R1 lt c4
1.00 X1 x(4-1) a4(R1 c(4-1)) 1.5
1.47(0.83-0.66) 1.75
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Discrete Distribution Inverse-transform
All discrete distributions can be generated by
the Inverse-transform technique.
F(x)
p1 p2 p3
p1 p2
R1
p1
General Form
b
a
c
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Discrete Distribution Inverse-transform

• Example Suppose the number of shipments, x, on
  the loading dock of IHW company is either 0, 1,
  or 2
• Data - Probability distribution
• Method - Given R, the generation
• scheme becomes

Consider R1 0.73 F(xi-1) lt R lt
F(xi) F(x0) lt 0.73 lt F(x1) Hence, x1 1
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Convolution Technique

• Use for X Y1 Y2 Yn
• Example of application
• Erlang distribution
• Generate samples for Y1 , Y2 , , Yn and then
  add these samples to get a sample of X.

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Erlang Distribution Convolution

• Example Generate 500 variates Xi with
  distribution Erlang-3 (mean k/????????
• Matlab Code
• for i1500,
• erlnum(i)-1/6(log(rand(1))log(rand(1))log(rand
  (1)))
• end

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Acceptance-Rejection technique

• Useful particularly when inverse cdf does not
  exist in closed form
• a.k.a. thinning
• Steps to generate X with pdf f(x)
• Step 0 Identify a majorizing function g(x) and a
  pdf h(x) satisfying
• Step 1 Generate Y with pdf h(x)
• Step 2 Generate U Uniform(0,1) independent of
  Y
• Step 3

Efficiency parameter is c
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Triangular Distribution Acceptance-Rejection

• Example

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Triangular Distribution Acceptance-Rejection
Matlab Code (for exactly 1000 samples) i0 while
ilt1000, Yrand(1) Urand(1) if
Ylt0.5 Ult2Y Ygt0.5 Ult2-2Y
ii1 X(i)Y end end
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Normal Distribution Special Technique

• Approach for normal(0,1)
• Consider two standard normal random variables, Z1
  and Z2, plotted as a point in the plane
• B2 Z21 Z22 chi-square distribution with 2
  degrees of freedom Exp(? 1/2). Hence,
• The radius B and angle ? are mutually
  independent.

??? Uniform(0,2??
In polar coordinates Z1 B cos ? Z2 B sin ?
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Normal Distribution Special Technique

• Approach for normal(?,??)
• Generate Zi N(0,1)
• Approach for lognormal(?,??)
• Generate X N(?,??)

Xi ? ? Zi
Yi eXi
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Normal Distribution Special Technique

• Generate 1000 samples of Normal(7,4)
• Matlab Code
• for i1500,
• R1rand(1)
• R2rand(1)
• Z(2i-1)sqrt(-2log(R1))cos(2piR2)
• Z(2i)sqrt(-2log(R1))sin(2piR2)
• end
• for i11000,
• Z(i)72Z(i)
• end

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