Chapter 1 Set Theory Flashcards
S
outcome space
outcome space is:
set of all possible events
disjoint
A intersect B = empty set
event
set of possible outcomes in a sample space
A={x imo S: x imo A} (imo=is a member of)
S is partitioned by B
If AiAj are pairwise disjoint AND U over all Ai = S, the collection Ai form a partition of S
Properties of Probability
Kolmogorov Axioms (for A imo B (Borel))
- P(B)>=0 for all A in B
- P(S)=1
- If Ai are imo B and pairwise disjoint, P(UAi)=sum P(Ai)
Probability of A^c=
P(A^c) = 1-P(A) (remember A and A^c partition S)
P(B and not A)=
P(BA^c) = P(B int A^c) = P(B)-P(B int A)
P(A U B) =
P(A OR B) = P(A) + P(B) - P(AB)
Probability of equally likely outcomes
P(A)=#A/#S
A U B (set notation)
{x: x is a member of A OR x is a member of B}
A intersect B (set notation)
{x: x is a member of A AND x is a member of B}
A^c (set notation)
{x: x is NOT a member of A}
associative prop of sets
AU(BUC)=(AUB)UC works for intersect as well, but NOT mixed
commutativity
AUB=BUA works for intersect as well
Distributive Laws
AU(BintC)=(AUB)int(AUC)
DeMorgans Laws
(AUB)^c = A^c int B^c
UAi (set notation)
{x is a member of S; x is a member of Ai for some i}
INT Ai (set notation)
{x is a member of S; x is a member of Ai for all i}
Sigma Algebra
A collection of subsets S is called a SA (B) IF:
a) 0 is a member of B
b) If A is a member of B, then A^c is an element of B
c) If Ai is a member of B, then U Ai is a member of B (closed)
Relation between Sigma Algebra and probability
Let S={s1,s2,..sn}. Let B be any SA of subsets of S. Let p1, p2, p3, … pn be nonnegative numbers that sum to 1. For any A in B, define P(A) = sum pi {i:si is a member of A)
Axiom of Finite Additivity
IF A and B are disjoint and imo SA, P(AUB)=P(A)+P(B)
P(BA^c) =
P(B)-P(AB)
P(AUB)=
P(A)+P(B)-P(AB)
If A is contained in B, then
P(A) le P(B)
Bonferroni’s Inequality
P(AB)>=P(A)+P(B)-1
P(A)= in terms of a partition set Ci
P(A)=sum(P(A int Ci)) if Ci’s partition A
A\B
part of A NOT in B {x:x imo A AND x !imo B}
P(A|B)=
P(A int B)/P(B)
Bayes’ Rule
P(A|B)=P(B|A)*P(A)/P(B)
P(A int B) =
P(A|B)P(B) OR
P(B|A)P(A)
P(A|B) = ? in terms of P(B|A)
P(B|A)*P(A) / P(B)
General Bayes’ Rule
P(Ai|B) = P(B|Ai)*P(Ai) / (sum [ P(B|Aj) * P(Aj) ] )
Two events are statistically independent IF
P(A int B) = P(A)P(B)
Probability is (descriptive words)
a real valued function mapping events to [0,1] that measures the randomness of an event
union
A U B = {x imo S: x imo A OR x imo B}
intersection
A int B = AB = {x imo S: x imo A AND x imo B}
compliment
A^c = {x imo S: x ! imo A}
{Ai} partitions S IIF
Ai int Aj = 0 for all i != j AND UAi=S for all i
Probability Space happens when
We have specified (S, B, P)
P(0) =
0
P(A^c) =
1-P(A)
P(AUB)=
P(A) + P(B) - P(A int B)
P(B int A^c)=
P(B) - P(AB)
P(A int B) >=
P(A) + P(B) -1 (Bonferroni’s inequality)
Domain of Probability
B (Sigma Algebra associated with P) {S, B, P)
Boole’s Inequality
P(UAi) le sum(P(Ai))
IF A subset B, P(A)
P(A) le P(B)
P(A|B) when A and B are disjoint
P(B|A)*P(A) / P(B) but P(A)=0; so P(A|B)=0
IF A and B are independent, so are the following:
A and B^c
A^c and B
A^c and B^c
Define probability
Given a sample space, S, with Sigma Algebra, B, a probability function measure is (any assigned) real-valued function with domain B that satisfies the Kolmogorov Axioms:
- Pr(A) ge 0
- Pr(S) = 1
- P(union Ai) = sum (P(Ai)) for any Ai in B, and Ai are mutually exclusive
Bayes P(A|X)=
P(X|A)P(A) / ( P(X|A)P(A) + P(X| not A)*P(not A) )
Chain rule
P(ABCD)=
P(A|BCD)P(BCD)
P(A|BCD)P(B|CD)P(CD)
P(A|BCD)P(B|CD)P(C|D)P(D)
