Set theory Flashcards
Sample space
set of all possible outcomes
Outcome
possible result of an experiment
Event
collection of possible outcomes of an experiment
A imo B therefor
x € A and
x € B
A = B therefor
A € B and
B € A
Union set representation
A U B = {x : x € A OR x € B}
Intersection set representation
A INT B = {x : x € A AND x € B}
Complement set representation
Ac = {x : x NOT imo A}
Elementary set operations
Communitativity A U B = B U A
B INT A = A INT B
Associativity = A U (B U C) = (A U B) U C
A INT (B INT C) = (A INT B) INT C
Distributive Laws A INT (B U C) = (A INT B) U (A INT C)
A U (B INT C) = (A U B) INT (A U C)
DeMorgan’s Laws (A U B)c = Ac INT Bc
(A INT B)c = Ac U Bc
Events A and B are disjoint IFF
A INT B = ø
Mutually exclusive
pairewise disjoint
Ai INT Bj = ø for all i ne j
Partition of S
If A1, A2, … are pairwise disjoint AND
Ui Ai = S (for all i)
then the collection Ai forms partition of S
Realization of an experiment
is an outcome in the sample space.
An event is a collection of possible outcomes.
A collection of subsets of S is called a Sigma Algebra (B) IFF
a. ø IMO B
b. If A IMO B, the Ac IMO B
c. If A1, A2, … IMO B, then Ui Ai IMO B
(B is closed under countable unions)
Axioms of Probability (Kolmogorov Axioms)
Given a sample space S and an associated Sigma Algebra B, a probability function is a function P with domain B that satisfies:
- P(A) ge 0
- P(S) = 1
- If A1, A2, … IMO B are pairwise disjoint, then
P(Ui) = Σ P(Ai) all i