MONTECARLO METHODS
PROBABILITY
CONDITIONAL PROBABILITY
INDIPENDENCE
DEFINITIONS
A PRIORI PROBABILITY of an EVENT
"It is defined as the # of cases in which the event can happen, over the total # of possible cases EQUALLY LIKELY"
A POSTERIORI PROBABILITY of an EVENT
"It is defined as the # of cases in which A occurs, over the total # of experiments in the SAME CONDITIONS"
AXIOMATIC PROBABILITY of an EVENT
"It is defined as a FUNCTION that has the EVENT SPACE AS DOMAIN and a 0<REAL #<1 AS CODOMAIN"
RANDOM PHENOMENON
"It is a PHENOMENON that OCCURS with a certain regularity, FOLLOWING A STATISTICAL LAW"
SAMPLE SPACE, S
"It is defined as A SET OF EVENTS containing ALL THE POSSIBLE OUTCOMES of a RANDOM PHENOMENON"
EVENT
"It represents ALL THE POSSIBLE COMBINATIONS of the ELEMENTS of the SAMPLE SPACE"
a SINGLE ELEMENT is called ELEMENTARY EVENT
EVENT SPACE, A
"It can be seen as a BOX containing ALL THE ELEMENTS (S) + ALL THEIR POSSIBLE COMBINATION + THE NULL EVENT"
KOLMOGOROV AXIOMS:
-
-
- when they are MUTUALLY EXCLUSIVE
TOTAL PROBABILITY FORMULA
A_1, A_2, A_3, A_4, A_5 sono cause dell'evento B
HYPOTHESIS
EXHAUSTIVE
BAYES FORMULA
DESCRIPTIVE STATISTICS
DEFINITIONS
RANDOM VARIABLE,
"It is a NON-INJECTIVE FUNCTION and associates a REAL NUMBER to EACH ELEMENTARY EVENT of S.
So it is a way to TRANSFORM an EVENT into a NUMBER and TRANSFER to it the PROBABILITY of the event itself"
MUTUALLY EXCLUSIVE
DISTRIBUTION of a RANDOM VARIABLE
"It is the DISTRIBUTION of PROBABILITY on the VALUES of "
DESCRETE
"If the distribution has a FINITE NUMBER OF VALUES"
CONTINUOUS
"If the distribution is CONTINUOUS on the R+ VALUES"
CUMULATIVE
PROPERTIES:
-
-
-
-
N.B. DISCONTINUITY is a SIGN of a DISCRETE R.V.
EXAMPLES:
PROBABILITY DENSITY FUNCTION
EXAMPLES: