ECE 730 Lectures 2 and 3 John A. Gubner UW-Madison ECE Dept. Jan. - - PowerPoint PPT Presentation

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ECE 730 Lectures 2 and 3 John A. Gubner UW-Madison ECE Dept. Jan. - - PowerPoint PPT Presentation

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ECE 730 Lectures 2 and 3 John A. Gubner UW-Madison ECE Dept. Jan. 26, 2009 Outline 1.4 Axioms and Properties of Probability Axioms Consequences of the Axioms 1.5 Conditional Probability The Law of Total Probability and Bayes Rule 1.4

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SLIDE 1

ECE 730 Lectures 2 and 3

John A. Gubner

UW-Madison ECE Dept.

  • Jan. 26, 2009
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SLIDE 2

Outline

1.4 Axioms and Properties of Probability Axioms Consequences of the Axioms 1.5 Conditional Probability The Law of Total Probability and Bayes’ Rule

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SLIDE 3

1.4 Axioms and Properties of Probability

Axioms

(i) The empty set ∅ is called the impossible event. P(∅) = 0. (ii) Probabilities are nonnegative; i.e., for any event A, P(A) ≥ 0. (iii) If A1, A2, . . . are events that are pairwise disjoint, then P ∞

  • n=1

An

  • =

∞

  • n=1

P(An). The technical term for this property is countable

  • additivity. In other words, “the probabilities of disjoint

events add.” (iv) The entire sample space Ω is called the sure event or the certain event, and its probability is one; i.e., P(Ω) = 1. If an event A = Ω satisfies P(A) = 1, we say that A is an almost-sure event.

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SLIDE 4

Consequences of the Axioms

Basic Results

◮ Finite Disjoint Unions.

P

  • N
  • n=1

An

  • =

N

  • n=1

P(An), An pairwise disjoint.

◮ Probability of a Complement (not compliment).

P(Ac) = 1 − P(A).

◮ Monotonicity. A ⊂ B

implies P(A) ≤ P(B). B A

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SLIDE 5

Consequences of the Axioms

Basic Results – continued

◮ Inclusion–Exclusion. P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

B A

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SLIDE 6

Consequences of the Axioms

Limit Results

◮ P

∞

  • n=1

An

  • = lim

N→∞ P

  • N
  • n=1

An

  • .

◮ P

∞

  • n=1

An

  • = lim

N→∞ P

  • N
  • n=1

An

  • .
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SLIDE 7

Consequences of the Axioms

Limit Results

◮ P

∞

  • n=1

An

  • = lim

N→∞ P

  • N
  • n=1

An

  • .

◮ P

∞

  • n=1

An

  • = lim

N→∞ P

  • N
  • n=1

An

  • .

( a )

A1 A2 A3

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SLIDE 8

Consequences of the Axioms

Limit Results

◮ P

∞

  • n=1

An

  • = lim

N→∞ P

  • N
  • n=1

An

  • .

◮ P

∞

  • n=1

An

  • = lim

N→∞ P

  • N
  • n=1

An

  • .

( a )

A1 A2 A3 A1 A2 A3

( b )

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SLIDE 9

Consequences of the Axioms

Limit Results – continued

( a )

A1 A2 A3 A1 A2 A3

( b )

◮ P

∞

  • n=1

An

  • = lim

N→∞ P(AN),

if An ⊂ An+1.

◮ P

∞

  • n=1

An

  • = lim

N→∞ P(AN),

if An+1 ⊂ An. These last two properties are called sequential continuity.

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SLIDE 10

Consequences of the Axioms

Limit Results – continued

◮ Union Bound or Countable Subadditivity.

P ∞

  • n=1

An

  • ≤

∞

  • n=1

P(An).

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SLIDE 11

1.5 Conditional Probability

Given two events A and B, P(A|B) := P(A ∩ B) P(B) . (1) This is equivalent to P(A ∩ B) = P(A|B)P(B). Interchanging the roles of A and B in (1) yields P(B|A) = P(A ∩ B) P(A) . It follows that P(B|A) = P(A|B)P(B) P(A) . In many cases, we are given P(B), P(A|B), P(Bc), and P(A|Bc), but we have to find P(A).

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The Law of Total Probability and Bayes’ Rule

Write A as the disjoint union A = (A ∩ B) ∪ (A ∩ Bc). Then P(A) = P(A ∩ B) + P(A ∩ Bc). We then have the Law of Total Probability, P(A) = P(A|B)P(B) + P(A|Bc)P(Bc). Substituting this last formula into P(B|A) = P(A|B)P(B) P(A) yields Bayes’ Rule, P(B|A) = P(A|B)P(B) P(A|B)P(B) + P(A|Bc)P(Bc).