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

  • 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

  • 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

Random variable is described by

Set of support

Probability measure

Probability measure is described by distribution function:


Probabilistic measure

Probabilistic measure

  • Probabilistic measure has three components:

    • Continuous;

    • Discrete (integer);

    • Singular.


Continuous r v

Continuous r.v.

Continuous r.v. is described by probability density function

Thus:


Continuous variable

Continuous variable

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


Discrete variable

Discrete variable

Discrete r.v. is described by mass probabilities:


Discrete variable1

Discrete variable

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


Singular variable

Singular variable

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 (Chebyshev, Kolmogorov)

hereare independent copies of r. v. ,


What did we learn

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 .


Centr al limit theorem gauss lindeberg

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

here


Beri es s en t h eorem

Beri-Essentheorem

where


What did we learn1

What did we learn ?

According to the LLN:

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


Example

Example

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

Statistical integrating …

???

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


Statistical integration

Statistical integration


Statistical simulation and monte carlo method

Statistical simulation and Monte-Carlo method

(Shapiro, (1985), etc)


Simulation of random variables

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


Basics of probability in statistical simulation and stochastic programming

N=100, 1000


Basics of probability in statistical simulation and stochastic programming

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


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