Probability Measures Flashcards
What is a probability measure (kolmogorovs axioms)
Probability measure (Kolmogorovs axioms) is:
Let omega be sample space and F a sigma-field on omega, mao P: F to [0,1] is a probability measure if:
P(omega) =1
Sigma additivity is satisfied
What is sigma-additivity
Sigma-additivity is:
If Fn n>=1 , is a collection of events which are pair wise disjoint (Fi N Fj = empty set mutually exclusive), then P(U n=1 to inf Fn) = sun n=1 to inf P(Fn)
What do kolmogorovs axioms imply
Kolmogorovs axioms imply that P(empty set)=0
1 = P(omega) = P(omega + empty set) (disjoint) = P(omega) + P(empty set)
What is a probability space
A probability space is the triplet (omega, F, P) (probability measure)
What is the fundamental model
Fundamental model is:
If omega = [0,1], consider B([0,1]) (generated by [0,x], for all x in [0,1])
What is the lebesgue measure on [0,1]
Lebesgue measure on [0,1] is:
Exists a unique probability measure on (omega, B[0,1]) s.t P[0,x]) = x for all x in [0,1]
P(U in [0,x]) = x if U is a uniform RV on [0,1]
What are properties of probability measures
Properties of probability measures are:
For all A in F, P(A^C) = 1 - P(A)
If A,B in F and A<= B then P(A) <= P(B)
A1,…,An are pairwise disjoint, then P(U I=1 to n Ai) = sum i=1 to n P(Ai)
Boole’s inequality
Inclusion exclusion (P(A U B) = P(A) + P(B) - P(A N B)
What is booles inequality/union bound
Booles inequality/union bound is:
For all A1,…An in F, we have P(U i=1 to n Ai) <= sum i=1 to n P(Ai)
What is inclusion-exclusion principle
Inclusion-exclusion principle is:
P(U i=1 to n Fi) = sum i=1 to n P(Fi) - sum i < j P(Fi N Fj) + sum i < j < k P(Fi N Fj N Fk)
= sum r=1 to n (-1)^(r-1) sum I1 < … < Ir P(Fi N…N Fir)
What is continuity of probability measures
Continuity of probability measures is:
Continuity from below
Continuity from above
What is continuity from below
Continuity from below is:
let An <= A(n+1) n >=1 and let A = U n=1 to inf An (sequence of non decreasing events)
P(A(n+1)) >= P(An) and lim n tends to inf P(An) = P(A)
What is continuity from above
Continuity from above is:
Let B(n+1) <= Bn and let B = N n=1 to inf Bn then P(B(n+1)) <= P(Bn) and lim n tends to inf P(Bn) = P(B)
