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
Basics of probability in statistical simulation and stochastic programming
Institute of Mathematics and Informatics
EURO Working Group on Continuous Optimization
Random variable is described by
Set of support
Probability measure is described by distribution function:
Continuous r.v. is described by probability density function
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. ,
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 .
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% )
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
Wrap-Up and conclusions