1. 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

2. 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

3. 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

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

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

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

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

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

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

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

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

12. 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

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

14. Beri-Essen theorem
where

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

16. 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: !!!

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

18. Statistical integration

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

20. 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

21. N=100, 1000

22. 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