Chapter 1 Set Theory

Question 1

S

Answer 1

outcome space

Question 2

outcome space is:

Answer 2

set of all possible events

Question 3

disjoint

Answer 3

A intersect B = empty set

Question 4

event

Answer 4

set of possible outcomes in a sample space

A={x imo S: x imo A} (imo=is a member of)

Question 5

S is partitioned by B

Answer 5

If AiAj are pairwise disjoint AND U over all Ai = S, the collection Ai form a partition of S

Question 6

Properties of Probability

Answer 6

Kolmogorov Axioms (for A imo B (Borel))

1. P(B)>=0 for all A in B
2. P(S)=1
3. If Ai are imo B and pairwise disjoint, P(UAi)=sum P(Ai)

Question 7

Probability of A^c=

Answer 7

P(A^c) = 1-P(A) (remember A and A^c partition S)

Question 8

P(B and not A)=

Answer 8

P(BA^c) = P(B int A^c) = P(B)-P(B int A)

Question 9

P(A U B) =

Answer 9

P(A OR B) = P(A) + P(B) - P(AB)

Question 10

Probability of equally likely outcomes

Answer 10

P(A)=#A/#S

Question 11

A U B (set notation)

Answer 11

{x: x is a member of A OR x is a member of B}

Question 12

A intersect B (set notation)

Answer 12

{x: x is a member of A AND x is a member of B}

Question 13

A^c (set notation)

Answer 13

{x: x is NOT a member of A}

Question 14

associative prop of sets

Answer 14

AU(BUC)=(AUB)UC works for intersect as well, but NOT mixed

Question 15

commutativity

Answer 15

AUB=BUA works for intersect as well

Question 16

Distributive Laws

Answer 16

AU(BintC)=(AUB)int(AUC)

Question 17

DeMorgans Laws

Answer 17

(AUB)^c = A^c int B^c

Question 18

UAi (set notation)

Answer 18

{x is a member of S; x is a member of Ai for some i}

Question 19

INT Ai (set notation)

Answer 19

{x is a member of S; x is a member of Ai for all i}

Question 20

Sigma Algebra

Answer 20

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)

Question 21

Relation between Sigma Algebra and probability

Answer 21

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)

Question 22

Axiom of Finite Additivity

Answer 22

IF A and B are disjoint and imo SA, P(AUB)=P(A)+P(B)

Question 23

P(BA^c) =

Answer 23

P(B)-P(AB)

Question 24

P(AUB)=

Answer 24

P(A)+P(B)-P(AB)

Question 25

If A is contained in B, then

Answer 25

P(A) le P(B)

Question 26

Bonferroni’s Inequality

Answer 26

P(AB)>=P(A)+P(B)-1

Question 27

P(A)= in terms of a partition set Ci

Answer 27

P(A)=sum(P(A int Ci)) if Ci’s partition A

Question 28

A\B

Answer 28

part of A NOT in B {x:x imo A AND x !imo B}

Question 29

P(A|B)=

Answer 29

P(A int B)/P(B)

Question 30

Bayes’ Rule

Answer 30

P(A|B)=P(B|A)*P(A)/P(B)

Question 31

P(A int B) =

Answer 31

P(A|B)P(B) OR

P(B|A)P(A)

Question 32

P(A|B) = ? in terms of P(B|A)

Answer 32

P(B|A)*P(A) / P(B)

Question 33

General Bayes’ Rule

Answer 33

P(Ai|B) = P(B|Ai)*P(Ai) / (sum [ P(B|Aj) * P(Aj) ] )

Question 34

Two events are statistically independent IF

Answer 34

P(A int B) = P(A)P(B)

Question 35

Probability is (descriptive words)

Answer 35

a real valued function mapping events to [0,1] that measures the randomness of an event

Question 36

union

Answer 36

A U B = {x imo S: x imo A OR x imo B}

Question 37

intersection

Answer 37

A int B = AB = {x imo S: x imo A AND x imo B}

Question 38

compliment

Answer 38

A^c = {x imo S: x ! imo A}

Question 39

{Ai} partitions S IIF

Answer 39

Ai int Aj = 0 for all i != j AND UAi=S for all i

Question 40

Probability Space happens when

Answer 40

We have specified (S, B, P)

Question 41

P(0) =

Answer 41

0

Question 42

P(A^c) =

Answer 42

1-P(A)

Question 43

P(AUB)=

Answer 43

P(A) + P(B) - P(A int B)

Question 44

P(B int A^c)=

Answer 44

P(B) - P(AB)

Question 45

P(A int B) >=

Answer 45

P(A) + P(B) -1 (Bonferroni’s inequality)

Question 46

Domain of Probability

Answer 46

B (Sigma Algebra associated with P) {S, B, P)

Question 47

Boole’s Inequality

Answer 47

P(UAi) le sum(P(Ai))

Question 48

IF A subset B, P(A)

Answer 48

P(A) le P(B)

Question 49

P(A|B) when A and B are disjoint

Answer 49

P(B|A)*P(A) / P(B) but P(A)=0; so P(A|B)=0

Question 50

IF A and B are independent, so are the following:

Answer 50

A and B^c
A^c and B
A^c and B^c

Question 51

Define probability

Answer 51

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:

1. Pr(A) ge 0
2. Pr(S) = 1
3. P(union Ai) = sum (P(Ai)) for any Ai in B, and Ai are mutually exclusive

Question 52

Bayes P(A|X)=

Answer 52

P(X|A)P(A) / ( P(X|A)P(A) + P(X| not A)*P(not A) )

Question 53

Chain rule

Answer 53

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)