5.3. Stochastic Modeling. Discrete RV

Random objects
Random events
Random variables
Random processes
Random point processes
BG
Transformation
α
X
Random objects and base generator

Discrete RV
METHODS OF GENERATION

Discrete RV given by distribution
A1
A2
…
Am
yes
no

Statistical processing for discrete RVs
Mathematical expectation (mean):
Variance:
ni is the number of appearances of value i (they are named as frequencies),
N is the total number of trials
Relative frequencies
Empiric expectation (average):
Empiric variance:
Absolute errors:
Relative errors:

Statistical processing for discrete RVs
Chi-squared test
Hypothesis that empiric distribution corresponds to the theoretical one is not true if and only if
α is a significance level

Laboratory #9
ASSIGNMENT:
Simulation of discrete random variable
Implement an algorithm for conducting a series of experiments to simulate a discrete random variable specified by the distribution
Calculate empirical probabilities, sample mean and variance, their relative errors
Calculate the chi-squared statistic and apply the chi-squared test for different values of N (N = 10, 100, 1,000, 10,000)
Draw a conclusion

Laboratory #9
ЗАДАНИЕ:
Имитационное моделирование дискретных случайных величин
Реализовать алгоритм проведения серии экспериментов по генерации дискретной случайной величины, заданной рядом распределения
Вычислить эмпирические вероятности, выборочные среднее и дисперсию, их относительные погрешности
Вычисление статистику хи-квадрат и применить критерий хи-квадрат при разных значениях N (N = 10, 100, 1 000, 10 000)
Сделать вывод

Laboratory #9
UI PROTOTYPE
Prob 1
Prob 2
Prob 3
Prob 4
Prob 5
Number of experiments
auto
Start
Average: 2.897 (error = 8%)
Variance: 2.072 (error = 9%)
Chi-squared: 13.51 > 9.488
is true

Uniform discrete distribution
Int is a truncating operation
FROM 0 TO n:
FROM a TO b:
Set n = b - a
Use
Calculate x = x + a
For example, if x from {1, 2, .., n} then use formula:
GENERATOR:

Geometric distribution
GENERATOR:

Negative binomial distribution
GENERATOR:

Binomial distribution
GENERATOR:
where
is a Heaviside step function

Poisson distribution

Poisson distribution generator
GENERATOR:

Laboratory #10
ASSIGNMENT:
Football manager game
Basketball

Laboratory #10
Use Poisson distribution for the number of goals in a match