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Probability Theory

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Probability Theory

Part 1: Basic Concepts

- Sample Point
- The outcome of a random experiment

- Sample Space S
- The set of all possible outcomes
- Discrete and Continuous

- Events
- A set of outcomes, thus a subset of S
- Certain, Impossible and Elementary

- Union
- Intersection
- Complement
- Properties
- Commutation
- Associativity
- Distribution
- De Morgan’s Rule

S

Axioms

If

If A1, A2, … are pairwise exclusive

Corollaries

- Sampling With Replacement and Ordering
- Sampling Without Replacement and With Ordering
- Permutations of n Distinct Objects
- Sampling Without Replacement and Ordering
- Sampling With Replacement and Without Ordering

- Conditional Probability of event A given that event B has occurred
- If B1, B2,…,Bn a partition of S, then
(Law of Total Probability)

S

B1

B2

A

B3

- If B1, …, Bn a partition of S then

Example

Which input is more probable if the output is 1? A priori, both input symbols are equally likely.

input

0

1

1-p

p

output

0

1

0

1

1-ε

ε

ε

1-ε

A

B

- Events A and B are independentif
- If two events have non-zero probability and are mutually exclusive, then they cannot be independent

1

1

½

½

C

1

½

1

1

½

½

1

Sequences of Independent Experiments

E1, E2, …, Ej experiments

A1, A2, …, Aj respective events

Independent if

Bernoulli Trials

Test whether an event A occurs (success – failure)

What is the probability of k successes in n independent repetitions of a Bernoulli trial?

Transmission over a channel with ε = 10-3 and with 3-bit majority vote