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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)

    10

  • 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

    15

    ][][]|[

    FPEFPFEP =

    joint probability

  • Example: binary communication system

    16

    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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