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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
2
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
master Management SPE 2011 - 2012 3
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)
4
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
master Management SPE 2011 - 2012 5
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.
master Management SPE 2011 - 2012 6
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.
master Management SPE 2011 - 2012 7
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:
master Management SPE 2011 - 2012 8
master Management SPE 2011 - 2012 9
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
master Management SPE 2011 - 2012 10
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).
master Management SPE 2011 - 2012 11
master Management SPE 2011 - 2012 12
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
master Management SPE 2011 - 2012 13
master Management SPE 2011 - 2012 14
Recapitulare elemente de statistica matematica
variabile aleatoare
master Management SPE 2011 - 2012 15
Recapitulare (2)
master Management SPE 2011 - 2012 16
Mecanismul de obtinere a v.a.