Generating Random Numbers

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Generating Random Numbers
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Generating Random Numbers
Anyone who considers arithmetical methods of
producing random digits is, of course, in a state
of sin. -- John von Neumann, 1951

• Generating truly random numbers is generally both
  impractical and in fact undesirable
• What we can do is generate pseudorandom numbers
• They can be reproduced starting from the same
  seed
• They must have good statistical properties (see
  RANDU for a remarkable failure)

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Uniform (0,1)

• Linear congruential method

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Methods

• Inverse Transform
• Acceptance-Rejection
• Convolution
• Composition

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Uniform

• Generate
• Return

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Uniform

• function genuni(N,a,b)
• urand(1,N)
• xa(b-a).u
• minxmin(x)
• maxxmax(x)
• NumBins51
• hhist(x,NumBins)
• for k1NumBins,
• bincenters(k)minx((maxx-minx)/NumBins)(k-1/
  2)
• end
• hh/sum(h)
• bar(bincenters,h)

Or use Matlab function unifrnd(a,b,M,N)
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Exponential

• Generate
• Return

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Exponential
function genexp(N,lambda) urand(1,N) x-1/lambda
.log(1-u)
Or use Matlab function exprnd(lambda,M,N)
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Normal

• Box-Muller Method
• Generate
• Generate
• Set
• Return
• Dont use it with adjacent numbers produced by a
  linear congruential generator

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Normal
function gennormal(N,mu,sigma) for i1N
u1rand u2rand z1sqrt(-2log(u1))cos(
2piu2) z2sqrt(-2log(u1))sin(2piu2)
x1(i) mu sigmaz1 x2(i) mu
sigmaz2 end
Or use Matlab function normrnd(mu,sigma,M,N)
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Binomial

• Generate IID Bernoulli(p) random
  numbers
• Return

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Binomial
function genbino(N,n,p) for i1N,
urand(1,n) y(ultp) x(i)sum(y) end
Or use Matlab function binornd(n,p,M,N)
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Geometric

• Generate
• Return

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Geometric
function gengeo(N,p) urand(1,N)
xceil(log(1-u)/log(1-p))
Or use Matlab function geornd(p,M,N)
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Poisson

• Set
• Generate and replace by
• If accept else increase
  by one and return to step 2

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Poisson
function genpois(N,lambda) for i1N,
k0p1 urand ppu while
pgtexp(-lambda) urand ppu
kk1 end x(i)k end
Or use Matlab function poissrnd(lambda,M,N)
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Multivariate Normal

• Generate IID random variables
• Decompose into a lower and upper triangular
  matrix
• (Cholesky decomposition)
• Return

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Multivariate Normal
Or use Matlab function mvnrnd(mu,sigma,N)