Transcript Sampling from Probability Distributions

Simulation Modeling and
Analysis
Sampling from
Probability Distributions
1
Outline
• Inverse Transforms for Random Variate
Generation
• Direct Transform and Convolution
• Acceptance-Rejection Technique
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Inverse Transforms for Random
Variate Generation
• Random variates are required to simulate
the vagaries of arrivals, service, processing
and the like which take place in the real
world.
• Once a reliable RNG is available, how does
one use it to obtain random variates with
selected statistical distributions?
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Steps in Inverse Transform Method
1.- Determine the cdf for the desired RV X.
2.- Set F(X) = R
3.- Solve F(X) = R for X in terms of R. I.e.
X = F-1(R)
4.- Generate the necessary RN sequence Ri
and use it to compute corresponding values
Xi for i = 1,2,…, n
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Inverse Transform Method for the
Exponential Distribution
1.- F(x) = 1 - e -  x , x > 0
2.- F(X) = 1 - e -  x = R
3.- X = - (1/) ln (1 - R)
4.- For i = 1,2,…, n
Xi = - (1/) ln (1 - Ri)
• Example: Use RAND in Excel to create 1000 PRN’s and
transform them to E. Examine your results in Stat::Fit.
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Inverse Transform Method for the
Uniform Distribution
1.- F(x) = {(x-a)/(b-a) ; 0 for a < x < b
2.- F(X) = (X -a)/(b-a) = R
3.- X = a + (b-a) R
4.- For i = 1,2,…, n
Xi = a + (b-a) Ri
• Example: Use RAND in Excel to create 1000 PRN’s and
transform them to U. Examine your results in Stat::Fit.
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Inverse Transform Method for the
Weibull Distribution
- x/)

1.- F(x) = 1 - e
, x>0

X/)
2.- F(X) = 1 - e
=R
3.- X =  [ ln (1 - R)]1/
4.- For i = 1,2,…, n
Xi =  [ ln (1 - Ri )]1/
• Example: Use RAND in Excel to create 1000 PRN’s and
transform them to W. Examine your results in Stat::Fit.
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Inverse Transform Method for the
Triangular Distribution
1.- F(x) = {0, for x <0; x2/2 for 0<x<1; 1-(2x)2/2, for 1 <x<2; 1, for x > 2.
2.- R=X2/2 (0<X<1) and R = 1- (2-X)2/2
(1<X<2)
3.- X = 2R1/2 , (0<R<1/2); X = 2 - (2(1-R))1/2 ,
(1/2<R<1)
• Example: Use RAND in Excel to create 1000 PRN’s and
transform them to W. Examine your results in Stat::Fit.
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Inverse Transform Method for
Empirical Distributions
•
•
•
•
Similar procedure applies
1.- Determine the empirical cdf F(x)
2.- Generate R
3.- Using the F(X) curve determine the
corresponding value of X.
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Transform Method for Distributions
without Closed Form Inverse
• Normal, gamma and beta distributions have
no closed form inverses. Inverse transform
method is applicable only approximately.
• For example, for the standard normal
distribution
X ~ (R0.135 - (1-R) 0.135)/0.1975
• Example: Use RAND in Excel to create 1000 PRN’s and
transform them to N. Examine your results in Stat::Fit.
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Inverse Transform Method for
Discrete Distributions
• Similar method applies.
– Lookup table
– Algebraic
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Direct Transform for the Normal
• For the normal distribution
x
F(x) = - (1/(2)1/2) e -(t2/2) dt
• Box-Muller method (pp. 341-343)
Z1 = (-2 ln R1)1/2 cos(2  R2)
Z2 = (-2 ln R1)1/2 sin(2  R2)
X1 =  +  Z1
X 2 =  +  Z2
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Convolution Method
• Convolution: the probability distribution of
two or more independent RV
• Useful for Erlang and binomial variates
• Erlang w/parameters (K,
– Sum of K independent exponential RV each
with mean 1/K
X = -(1/ K ) ln  Ri
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Acceptance-Rejection Technique
• Say random variates X are needed
uniformly distributed between 1/4 and 1
• Steps
1.- Generate R
2a.- If R > 1/4 make X = R and go to 3
2b.- If R < 1/4 go to 1 until X is obtained
3.- Repeat from 1 for another X
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Acceptance-Rejection Technique for
Poisson Distribution
• Recall Poisson and exponential are closely
related.
P(N=n) = e   n/n!
• Steps
1.- Set n =0, P=1
2.- Generate R and replace P by P R
3.- If P < e , accept N = n, otherwise go to 2
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Acceptance-Rejection Technique for
Gamma Distribution
• Steps
1.- Compute a = (2-1)1/2 ; b = 2 - ln 4 + 1/a
2.- Generate R1, R2
3.- Compute X =  [R1/(1-R1)]a
4a.- If X > b-ln(R12R2) reject and go to 2
4b.- If X < b-ln(R12R2) use X as is or as X/
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