Set theory Flashcards

1
Q

Sample space

A

set of all possible outcomes

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2
Q

Outcome

A

possible result of an experiment

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3
Q

Event

A

collection of possible outcomes of an experiment

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4
Q

A imo B therefor

A

x € A and

x € B

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5
Q

A = B therefor

A

A € B and

B € A

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6
Q

Union set representation

A

A U B = {x : x € A OR x € B}

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7
Q

Intersection set representation

A

A INT B = {x : x € A AND x € B}

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8
Q

Complement set representation

A

Ac = {x : x NOT imo A}

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9
Q

Elementary set operations

A

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

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10
Q

Events A and B are disjoint IFF

A

A INT B = ø

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11
Q

Mutually exclusive

A

pairewise disjoint

Ai INT Bj = ø for all i ne j

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12
Q

Partition of S

A

If A1, A2, … are pairwise disjoint AND

Ui Ai = S (for all i)

then the collection Ai forms partition of S

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13
Q

Realization of an experiment

A

is an outcome in the sample space.

An event is a collection of possible outcomes.

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14
Q

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

A

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)

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15
Q

Axioms of Probability (Kolmogorov Axioms)

A

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(Ui) = Σ P(Ai) all i

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16
Q

Properties of probability

A
  1. P(ø) = 0
  2. P(A) le 1
  3. P(Ac) = 1 - P(A)
17
Q

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

A
  1. P(B INT Ac) = 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)
18
Q

Bonferroni Inequality

A

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

19
Q

Boole’s inequality

A

P(U Ai ) le Σ P(Ai )

20
Q

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

A

P(A) = Σ ( INT Ci )

21
Q

Fundamental theorem of counting

A

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

22
Q

statistically independent

A

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

23
Q
A
24
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A
25
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26
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27
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28
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29
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30
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A