Lecture 2 Basics of probability in statistical simulation and stochastic programming Leonidas Sakalauskas Institute of Mathematics and Informatics Vilnius, Lithuania EURO Working Group on Continuous Optimization

Content Random variables and random functions Law of Large numbers Central Limit Theorem Computer simulation of random numbers Estimation of multivariate integrals by the Monte-Carlo method

Simple remark Probability theory displays the library of mathematical probabilistic models Statistics gives us the manual how to choose the probabilistic model coherent with collected data Statistical simulation (Monte-Carlo method) gives us knowledge how to simulate random environment by computer

Random variable Random variable is described by Set of support Probability measure Probability measure is described by distribution function:

Probabilistic measure Probabilistic measure has three components: Continuous; Discrete (integer); Singular.

Continuous r.v. Continuous r.v. is described by probability density function Thus:

Continuous variable If probability measure is absolutely continuous, the expected value of random function:

Discrete variable Discrete r.v. is described by mass probabilities:

Discrete variable If probability measure is discrete, the expected value of random function is sum or series:

Singular variable Singular r.v. probabilistic measure is concentrated on the set having zero Borel measure (say, Kantor set).

Law of Large Numbers (Chebyshev, Kolmogorov) here are independent copies of r. v. ,

What did we learn ? The integral is approximated by the sampling average if the sample size N is large, here is the sample of copies of r.v. , distributed with the density .

Central limit theorem (Gauss, Lindeberg, ...) here

Beri-Essen theorem where

What did we learn ? According to the LLN: Thus, apply CLT to evaluate the statistical error of approximation and its validity.

Example Let some event occurred n times repeating N independent experiments. Then confidence interval of probability of event : (1,96 – 0,975 quantile of normal distribution, confidence interval – 5% ) here If the Beri-Esseen condition is valid: !!!

Statistical integrating … ??? Main idea – to use the gaming of a large number of random events

Statistical integration

Statistical simulation and Monte-Carlo method (Shapiro, (1985), etc)

Simulation of random variables There is a lot of techniques and methods to simulate r.v. Let r.v. be uniformly distributed in the interval (0,1] Then, the random variable , where , is distributed with the cumulative distribution function

N=100, 1000

Wrap-Up and conclusions the expectations of random functions, defined by the multivariate integrals, can be approximated by sampling averages according to the LLN, if the sample size is sufficiently large; the CLT can be applied to evaluate the reliability and statistical error of this approximation