# introduction to random processes (1): probability ?· 01-02-2012 · introduction to random...

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Introduction to Random Processes (1): Probability

Luiz DaSilva Professor of Telecommunications dasilval@tcd.ie +353-1-8963660

Brief review of set theory: a set q A set is a collection of elements

q May be finite, or countably or uncountably infinite

q Examples

qA finite set:

qA countably infinite set:

qAn uncountably infinite set

2

Question q Consider an N-element set

q How many subsets of this set are there?

3

Set algebra q Union

q Intersection

q Complement

q Difference (reduction)

4

Set algebra example A = {odd integers between 0 and 10} B = {multiples of 3 between 1 and 10} A B = A U B = A \ B =

5

Disjoint and equal sets q Two sets are disjoint if:

q Two sets are equal if:

6

De Morgans Laws

7

Random experiments qA random experiment H is an experiment for

which the outcome is not known a priori

qThe sample space is the set of all possible outcomes of H

qAn event E is a subset of qThe null set denotes the impossible event

8

Example A random experiment:

H = roll a fair die once

The sample space: An event:

9

Probability Defn: Probability is a set function P[] that assigns to each event E in a -field F a number P[E] such that (i) (ii) (iii)

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Axiom #1 P[] = 0 Proof:

11

Axiom #2 P[E] = 1 P[Ec] Proof:

12

Axiom #3 When Eis are disjoint: Proof: [A special case of property (iii) in the definition]

13

][][11==

=n

ii

n

ii EPEP

Axiom #4 P[E U F] = P[E] + P[F] P[EF] Proof:

14

Conditional probability Defn: The probability of event E conditional on event F, with P[F] > 0, is Interpretation: Provability of event E given that event F has occurred

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][][]|[

FPEFPFEP =

joint probability

Example: binary communication system

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X Y

noisy channel

0

1

0

1

1 -

1 -

Theorem on total probability Thm: Let E1, , En be disjoint and exhaustive (their union yields ), with P[Ei] 0. Then, for any event B, Proof:

17

][]|[][1

i

n

ii EPEBPBP

=

=

Bayes theorem Thm: Let Eis be a partition of (disjoint, exhaustive), with P[Ei] 0. Then, for any event B, with P[B] 0,

18

=

== n

jjj

iiii

EPEBP

EPEBPBPBEPBEP

1][]|[

][]|[][][]|[

Binary channel example revisited q If receiver decodes a 1, what is the probability

that a 1 was sent?

19

Independence Defn: (i) Events E and F are independent if P[EF] = P[E]P[F] (ii) Events E1, , En are independent if

20

][]...[

][][][][

][][][

121

=

=

=

=

n

iin

kjikji

jiji

EPEEEP

kjiEPEPEPEEEPjiEPEPEEP

!

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