Postscript version of these notes

STAT 380 Lecture 24

Simulation

Monte Carlo computation of expected value:

To compute : do experiment n times to generate n independent observations all with same distribution as X.

Get .

Use

as an estimate of

To estimate : use same and compute

Random Number Generation

In practice: random quantities are not used.

Use: pseudo-random uniform random numbers.

They ``behave like'' iid Uniform(0,1) variables.

Many, many generators. One standard kind: linear congruential generators.

Start with an integer in range . Compute

Here the mod means compute the remainder after division by m.

Integers a and b are chosen so that

• the sequence runs through the set (or as much of it is possible, sometimes).
• the sequence of pairs

  

  fills up as much of the square .

• similarly for triples .

Use

as Uniform(0,1) random numbers.

General Random Variables

We now pretend we can generate which are iid Uniform(0,1).

Monte Carlo methods for generating a sample:

• Transformation:
  • Inverse Probability Integral Transform
  • Multivariate (e.g. Box Müller)

• Acceptance Rejection
• Markov Chain Monte Carlo