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. Content. Random variables and random functions Law of Large numbers

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Basics of probability in statistical simulation and stochastic programming

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


Content
Content stochastic programming

  • 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
Simple remark stochastic programming

  • 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 stochastic programming

Random variable is described by

Set of support

Probability measure

Probability measure is described by distribution function:


Probabilistic measure
Probabilistic measure stochastic programming

  • Probabilistic measure has three components:

    • Continuous;

    • Discrete (integer);

    • Singular.


Continuous r v
Continuous r.v. stochastic programming

Continuous r.v. is described by probability density function

Thus:


Continuous variable
Continuous variable stochastic programming

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


Discrete variable
Discrete variable stochastic programming

Discrete r.v. is described by mass probabilities:


Discrete variable1
Discrete variable stochastic programming

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


Singular variable
Singular variable stochastic programming

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


Law of large numbers ch eby sh ev kolmogorov
Law of Large Numbers stochastic programming (Chebyshev, Kolmogorov)

hereare independent copies of r. v. ,


What did we learn
What did we learn ? stochastic programming

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 .


Centr al limit theorem gauss lindeberg
Centr stochastic programmingal limit theorem (Gauss, Lindeberg, ...)

here


Beri es s en t h eorem
Beri-Es stochastic programmingsentheorem

where


What did we learn1
What did we learn ? stochastic programming

According to the LLN:

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


Example
Example stochastic programming

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: !!!


Statisti cal integra t i ng
Statisti stochastic programmingcal integrating …

???

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


Statistical integration
Statistical integration stochastic programming


Statistical simulation and monte carlo method
Statistical simulation and Monte-Carlo method stochastic programming

(Shapiro, (1985), etc)


Simulation of random variables
Simulation of random variables stochastic programming

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 stochastic programming=100, 1000


Wrap-Up and conclusions stochastic programming

  • 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


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