Set theory

Question 1

Sample space

Answer 1

set of all possible outcomes

Question 2

Outcome

Answer 2

possible result of an experiment

Question 3

Event

Answer 3

collection of possible outcomes of an experiment

Question 4

A imo B therefor

Answer 4

x € A and

x € B

Question 5

A = B therefor

Answer 5

A € B and

B € A

Question 6

Union set representation

Answer 6

A U B = {x : x € A OR x € B}

Question 7

Intersection set representation

Answer 7

A INT B = {x : x € A AND x € B}

Question 8

Complement set representation

Answer 8

A^c = {x : x NOT imo A}

Question 9

Elementary set operations

Answer 9

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 = A^c INT B^c

(A INT B)^c = A^c U B^c

Question 10

Events A and B are disjoint IFF

Answer 10

A INT B = ø

Question 11

Mutually exclusive

Answer 11

pairewise disjoint

A_i INT B_j = ø for all i ne j

Question 12

Partition of S

Answer 12

If A₁, A₂, … are pairwise disjoint AND

U_i A_i = S (for all i)

then the collection A_i forms partition of S

Question 13

Realization of an experiment

Answer 13

is an outcome in the sample space.

An event is a collection of possible outcomes.

Question 14

A collection of subsets of S is called a Sigma Algebra (B) IFF

Answer 14

a. ø IMO B
b. If A IMO B, the A^c IMO B
c. If A₁, A₂, … IMO B, then U_i A_i IMO B

(B is closed under countable unions)

Question 15

Axioms of Probability (Kolmogorov Axioms)

Answer 15

Given a sample space S and an associated Sigma Algebra B, a probability function is a function P with domain B that satisfies:

1. P(A) ge 0
2. P(S) = 1
3. If A₁, A₂, … IMO B are pairwise disjoint, then

P(U_i) = Σ P(A_i) all i

Question 16

Properties of probability

Answer 16

1. P(ø) = 0
2. P(A) le 1
3. P(A^c) = 1 - P(A)

Question 17

Relations between P(A) and P(B) iff A and B are any sets in Borel

Answer 17

1. P(B INT A^c) = P(B) - P(A INT B)
2. P(A U B) = P(A) + P(B) - P(A INT B)
3. If A subset B. then P(A) le P(B)

Question 18

Bonferroni Inequality

Answer 18

P(A INT B) ge P(A) + P(B) -1

Question 19

Boole’s inequality

Answer 19

P(U A_i ) le Σ P(A_i )

Question 20

For a set of events A and a set of partitions of A, C_i, P(A) =

Answer 20

P(A) = Σ ( INT C_i )

Question 21

Fundamental theorem of counting

Answer 21

If a job consists of k tasks, the ith of which can be done in n_i ways, then the entire job can be done in n₁ * … * n_k ways.

Question 22

statistically independent

Answer 22

P(A INT B) = P(A) P(B)

Question 23

Answer 23

Question 24

Question 25

Answer 25

Question 26

Answer 26

Question 27

Answer 27

Question 28

Answer 28

Question 29

Answer 29

Question 30

Answer 30