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Master Management 2011-2012

Simularea proceselor economice

- Curs 3

Metoda Monte Carlo

cu EXCEL n cazul

variabilelor probabiliste

discrete si continue

master Management SPE 2011 - 2012

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CUPRINS I. Procedura pentru aplicarea metodei Monte Carlo n

cazul variabilelor probabiliste discrete

D i discrete de probabilitate

Poisson .

II. Procedura pentru aplicarea metodei Monte Carlo n

cazul variabilelor probabiliste continue

D i continue de probabilitate continu

continu

triunghiular

normal

exponential master Management SPE 2011 - 2012

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v. a. discrete vs. continue

O v.a. este o cantitate masurata in legatura cu un experiment aleator;

O sau o nu este altceva dect un

alt mod de a descrie rezultatul unui experiment aleator. V.a sunt importante deoarece asigura obiectivitate in reproducerea

/replicarea unor rezultate ale unor evenimente (prin verificabilitate, reproductibilitate) . Analistul/decidentul/cercetatorul alege procentajul din masa de evenimente replicate care ar trebui in principiu sa conduca la rezultate similare (sa fie acceptate in baza formularii ipotezei verificate ca fiind CORECTA, si sa respinga ipoteza FALSA), permitand variabilitatea rezultatelor.

Nivel de incredere: 1 = 0.90, 0.95, 0.99

Nivel de semnificatie: = 0.10, 0.05, 0.01(erori acceptate)

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Clasificare v.a.

Numerice

Discrete vs. continue

Categoriale

Nominale/ordinale Ex1: succes/esec

clase: I, II, III etc. sau atribute calitative

master Management SPE 2011 - 2012

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Descrierea v.a.

f(x) functie de masa/densitate de probabilitate

F(x) functie de repartitie

Indicatori statistici:

medie,

dispersie, abatere standard

etc.

Terminology - Probability mass function a probability mass function (pmf) is a function that gives the probability that a discrete random variable

is exactly equal to some value. A pmf differs from a probability density function (pdf) in that the values

of a pdf, defined only for continuous random variables, are not probabilities as such. Instead, the integral

of a pdf over a range of possible values (a, b] gives the probability of the random variable falling within

that range.

Suppose that X: S R is a discrete random variable defined on a sample space S. Then the probability mass

function fX: R [0, 1] for X is defined as

Note that fX is defined for all real numbers, including those not in the image of X; indeed, fX(x) = 0 for all x

X(S).

Since the image of X is countable, the probability mass function fX(x) is zero for all but a countable number of

values of x.

The discontinuity of probability mass functions reflects the fact that the cumulative distribution function of a

discrete random variable is also discontinuous. Where it is differentiable, the derivative is zero, just as the

probability mass function is zero at all such points.

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Probability density function

a probability density function

(abbreviated as pdf, or just density)

of an absolutely continuous random

variable is a function that describes

the relative chance for this random

variable to occur at a given point in

the observation space. The

probability for a random variable to

fall within a given set is given by

the integral of its density over the

set.

The terms probability distribution

function and probability

function have also been used to

denote the probability density

function.

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Cumulative distribution function

The cumulative distribution function (CDF), or just

distribution function, describes the probability that a real-

valued random variable X with a given probability

distribution will be found at a value less than or equal to

x.

Intuitively, it is the "area so far" function of the probability

distribution.

For every real number x, the CDF of a real-valued random

variable X is given by

where the right-hand side represents the probability that the

random variable X takes on a value less than or equal to

x.The probability that X lies in the interval (a, b] is therefore

FX(b) FX(a) if a < b.

The CDF of X can be defined in terms of the probability density

function as follows:

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Exemple de v.a.

Distributie Parametri

Empirica

Se folosesc numere aleatoare

Normala

Media m si abaterea standard s

Uniforma a si b

Exponentiala

b

Triangulara

a, m, si b

Weibull

a si b

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Distributii simetrice

Corespund unor valori situate simetric fata de o valoare centrata

Daca exista valori deviante (outliers), distributie devine asimetrica

(skewed), formand eventual coada (tail).

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Pentru variabile discrete

O functie de probabilitate discreta

- este un tabel, grafic sau regula care arata toate valorile unei

v.a. discrete X si probabilitatile lor corespunzatoare.

Orice functie de probabilitate discreta satisface urmatoarele reguli:

- nici o probabilitate nu poate fi negativa: P(X=x) 0

- suma tuturor probabilitatilor este 1: P(X=x1)+P(X=x2)+ =1

Functia cumulativa de probabilitate arata probabilitatea ca X

sa ia o valoare cu o valoare particulara data:

F(X=x)=P(X x)

Proprietati: 0 F(b) 1 pentru oricare b

daca a

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Recapitulare elemente de statistica matematica

variabile aleatoare

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Recapitulare (2)

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Mecanismul de obtinere a v.a.