011_02_IntroToSimulation

Core ideas in Monte Carlo simulation

modular arithmetic gives pseudo-random streams that are indistiguishable from 'true' Uniformly distributed samples in integers from $\{0,1,2,...,m\}$
by diving the integer streams from above by $m$ we get samples from $\{0/m,1/m,...,(m-1)/m\}$ and "pretend" this to be samples from the Uniform(0,1) RV
we can use inverse distribution function of von Neumann's rejection sampler to convert samples from Uniform(0,1) RV to the following:
- any other random variable
- vector of random variables that could be dependent
- or more generally other random structures:
  - random graphs and networks
  - random walks or (sensible perturbations of live traffic data on open street maps for hypothesis tests)
  - models of interacting paticle systems in ecology / chemcal physics, etc...

import breeze.stats.distributions._ val poi = new Poisson(3.0);

import breeze.stats.distributions._ poi: breeze.stats.distributions.Poisson = Poisson(3.0)

val s = poi.sample(5); // let's draw five samples - black-box

s: IndexedSeq[Int] = Vector(2, 1, 1, 3, 2)

s.map( x => poi.probabilityOf(x) ) // PMF

res0: IndexedSeq[Double] = Vector(0.22404180765538775, 0.14936120510359185, 0.14936120510359185, 0.22404180765538775, 0.22404180765538775)

val doublePoi = for(x <- poi) yield x.toDouble // meanAndVariance requires doubles, but Poisson samples over Ints

doublePoi: breeze.stats.distributions.Rand[Double] = MappedRand(Poisson(3.0),$Lambda$5455/238879880@20291ada)

breeze.stats.meanAndVariance(doublePoi.samples.take(1000));

res1: breeze.stats.meanAndVariance.MeanAndVariance = MeanAndVariance(2.9940000000000055,3.0370010010010025,1000)

(poi.mean, poi.variance) // population mean and variance

res2: (Double, Double) = (3.0,3.0)