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products/sources/formale Sprachen/PVS/probability/ (Beweissystem der NASA Version 6.0.9^©) Datei vom 28.9.2014 mit Größe 3 kB

Quelle probability_space.pvs Sprache: PVS

probability_space[T:TYPE+,(IMPORTING measure_integration@subset_algebra_def[T])
                  S:sigma_algebra,   (IMPORTING probability_measure[T,S])
                  P:probability_measure]: THEORY

BEGIN

   IMPORTING measure_integration@sigma_algebra[T,S],
             probability_measure[T,S],
             continuous_functions_aux[real],
             measure_integration@measure_space[T,S],
             measure_integration@measure_props[T,S,to_measure(P)],
             measure_integration@real_borel

   limit: MACRO [(convergence_sequences.convergent?)->real]
                                                  = convergence_sequences.limit

   phi,g,h: VAR borel_function
   B1,B2:   VAR borel
   A,B: VAR (S)
   x,y: VAR real
   n0z: VAR nzreal
   t:   VAR T
   n:   VAR nat
   X,Y: VAR random_variable
   XS:  VAR [nat->random_variable]

   null?(A)         :bool = P(A) = 0
   non_null?(A)     :bool = NOT null?(A)
   independent?(A,B):bool = P(intersection(A,B)) = P(A) * P(B)

% Note carefully that the above DOES NOT say: ... = 0

   zero: random_variable = (LAMBDA t: 0)
   one:  random_variable = (LAMBDA t: 1)

   ; % needed for syntax purposes!

   <=(X,x):(S) = {t | X(t) <= x};
   <(x,X): (S) = {t | X(t) >  x};
   =(X,x): (S) = {t | X(t) =  x};
   <(X,x): (S) = {t | X(t) <  x};
   <=(x,X):(S) = {t | X(t) >= x};
   /=(X,x):(S) = {t | X(t) /= x};

   complement_le1: LEMMA complement(X <= x) = (x <  X)
   complement_lt1: LEMMA complement(x <  X) = (X <= x)
   complement_eq : LEMMA complement(X =  x) = (X /= x)
   complement_lt2: LEMMA complement(X <  x) = (x <= X)
   complement_le2: LEMMA complement(x <= X) = (X <  x)
   complement_ne:  LEMMA complement(X /= x) = (X =  x)

   ; % needed for syntax purposes!

   +(X,x)  :random_variable = (LAMBDA t: X(t) + x);
   +(x,X)  :random_variable = (LAMBDA t: x + X(t));
   -(X,x)  :random_variable = (LAMBDA t: X(t) - x);
   -(x,X)  :random_variable = (LAMBDA t: x - X(t));
   /(X,n0z):random_variable = (LAMBDA t: X(t)/n0z);

   independent?(X,Y):bool
     = FORALL B1,B2: independent?(inverse_image(X,B1),inverse_image(Y,B2))

   borel_comp_rv_is_rv: JUDGEMENT o(phi,X) HAS_TYPE random_variable

   borel_independence: LEMMA independent?(X,Y) => independent?(g o X, h o Y)

   partial_sum_is_random_variable:
     LEMMA random_variable?(LAMBDA t: sigma(0,n,LAMBDA n: XS(n)(t)))

   distribution_function?(F:[real->probability]):bool
                                       = EXISTS X: FORALL x: F(x) = P(X <= x)

   distribution_function: TYPE+ = (distribution_function?) CONTAINING
                                      (LAMBDA x: IF x < 0 THEN 0 ELSE 1 ENDIF)

   distribution_function(X)(x):probability = P(X <= x)

   convergence_in_distribution?(XS,X):bool
     = FORALL x: continuous?(distribution_function(X),x) IMPLIES
                  convergence((LAMBDA n: distribution_function(XS(n))(x)),
                                         distribution_function(X)(x))
% Lemma 2.1.11 (G&S)

   invert_distribution:   LEMMA LET F = distribution_function(X) IN
                                P(x < X) = 1 - F(x)
   interval_distribution: LEMMA LET F = distribution_function(X) IN
                                x <= y IMPLIES
                                P(intersection(x < X, X <= y)) = F(y) - F(x)
   limit_distribution:    LEMMA LET F = distribution_function(X) IN
                                P(X = x) = F(x) - limit(LAMBDA n: F(x-1/(n+1)))

% Lemma 2.1.6 in G&S

   F: VAR distribution_function

   distribution_0:                LEMMA convergence(F o (lambda (n:nat): -n),0)
   distribution_1:                LEMMA convergence(F,1)
   distribution_increasing:       LEMMA increasing?[real](F)
   distribution_right_continuous: LEMMA right_continuous?(F)


END probability_space

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