Basics of probability in statistical simulation and stochastic programming

Leonidas SakalauskasInstitute of Mathematics and InformaticsVilnius, LithuaniaEURO Working Group on Continuous Optimization

Lecture 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

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:

)(Pr)( xXobxF

)(XSUPP

Probabilistic measure

Probabilistic measure has three components:

Continuous;Discrete (integer);Singular.

Continuous r.v.

Continuous r.v. is described by probability density function )(xf

Thus:

x

dyyfxF )()(

Continuous variable

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

dxxpxfXEf )()()(

Discrete variable

Discrete r.v. is described by mass probabilities:

n

n

ppp

xxx

,...,,

,...,,

21

21

Discrete variable

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

n

iii pxfXEf

1

)()(

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)

,lim 1 zN

zN

i i

N

here are independent copies of r. v. ,

Ez

Nzzz ,...,, 21

What did we learn ?

N

N

i i 1

is the sample of copies of r.v. , distributed with the density .

Nzzz ,...,, 21

dzzpzxf )(),(The integral

is approximated by the sampling average

,,...,1),,( Njzxf jj if the sample size N is large, here

)(zp

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

),(/

lim xxN

xP N

N

,2

1)( 2

2

x y

dyex

here

,EX 222 )( XEXD,1

N

x

x

N

ii

N

Beri-Essen theorem

N

XExxFN

x

3

3

41.0)()(sup

xxobxF NN Pr)(where

What did we learn ?

According to the LLN:

,1

1

N

iixN

x,

)(1

2

2

N

xxN

iNi

N

xx

EXXE

N

iNi

1

3

3

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 :

N

ppp

N

ppp

)1(96.1,

)1(96.1

here ,N

np (1,96 – 0,975 quantile of normal distribution,

confidence interval – 5% )

6)1( ppNIf the Beri-Esseen condition is valid: !!!

Statistical integrating …

b

a

dxxfI )( ???

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

Statistical integration

dxxpxfXEf )()()(

,)(

1

N

xfNi i )(pxi

Statistical simulation and Monte-Carlo method

x

dzzpzxfxF min)(),()(

,min),(

1

x

N

i i

N

zxf )(pzi

(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 ,

)(UFU

is distributed with the cumulative distribution function )(F

))sin(cos()( xaxxxf

0

))sin(cos()( dxexaxxaF x

N=100, 1000

Wrap-Up and conclusions

o 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;

o the CLT can be applied to evaluate the reliability and statistical error of this approximation