Basics of probability in statistical simulation and stochastic programming

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Basics of probability in statistical simulation and stochastic programming

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Basics of probability in statistical simulation and stochastic programming

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Basics of probability in statistical simulation and stochastic programming

Lecture 2

Leonidas Sakalauskas

Institute of Mathematics and Informatics

Vilnius, Lithuania

EURO Working Group on Continuous Optimization

- 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

- 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 is described by

Set of support

Probability measure

Probability measure is described by distribution function:

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

Continuous r.v. is described by probability density function

Thus:

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

Discrete r.v. is described by mass probabilities:

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

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

hereare independent copies of r. v. ,

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 .

here

where

According to the LLN:

Thus, apply CLT to evaluate the statistical error of approximation and its validity.

Let some event occurred ntimes repeating Nindependent 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: !!!

???

Main idea – to use the gaming of a large number of random events

(Shapiro, (1985), etc)

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