Unit-1_ Lec 1_ ITA

Embed Size (px)

Citation preview

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    1/32

    Review of Basic Probability

    Unit-1

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    2/32

    What is Probability?

    Deterministic phenomena

    Daily sunrises andsunsets

    Tides at sea shores

    Phases of the moon

    Seasonal changes in

    weather

    Annual flooding of theYamuna

    Random phenomena

    Results of coin tosses

    Results of rolling dice

    Results of horse races

    World refer to dice games

    and gambling

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    3/32

    Probabilistic notions are common placein everyday language usage

    We use words such as

    Probable/improbable; possible/impossible

    Certain/uncertain; likely/unlikely

    Phrases such as there is a 50-50 chance

    The probability of precipitation is 20%

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    4/32

    PROBABILITY ?

    The theory of probability deals with averagesof mass phenomenon occurring sequentially orsimultaneously.

    The purpose of theory is to describe andpredict averages in terms of probabilities ofevents.

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    5/32

    Definition

    Three Aproaches to Probability:

    Classical

    Relative frequency Axiomatic

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    6/32

    Classical (Priori) Approach to Probability:

    The probability of Heads is because there are twosides to the coin

    This is called the classical approach to probability

    More generally, if there are n possible outcomes of anexperiment, then each outcome has probability 1/n

    Justification: Symmetry principle (Equally likely);

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    7/32

    Problems with Classical Approach:

    What exactly is an outcome?If we toss two coins, are there three outcomes or fouroutcomes?

    {0 Heads, 1Head, 2 Heads}?

    {(T,T), ( T,H), ( H,T), ( H,H)}?

    Note that 2 Heads has probability 1/3 or dependingon the choice

    There are only two outcomes: either I Win theLottery, or I dont, so probability is 1/2?

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    8/32

    Relative Frequency (Posteriori) Approach:

    the probability of Heads is because when tossed, thecoin will turn up Heads half the time

    How do we know the coin will turn up Heads half the

    time? Suppose multiple heads tosses have resulted in 50%

    Heads.

    Setting P (Head)=1/2 is the relative frequency approach

    to probability

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    9/32

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    10/32

    Relative Frequency (Posteriori)Approach contd

    If an outcome x occurs m times on N trails, its

    relative frequency is m/N & we define itsprobability P(x) to be m/N

    Does there exist a probability of Heads for newunbiased untossed coin?

    Or do probabilities come into existence only aftermultiple tosses?

    How large N should be?

    Are probabilities re-defined after each toss?

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    11/32

    Probability as beliefs:

    Many assertions about probability are

    essentially statements of beliefs A fair coin is one for which P(Heads)=1/2

    but how do we know whether a given coin

    is fair? Symmetry of the physical object is a belief

    That further tosses of a coin for which

    P(Heads) =1/2 will result in 50% Heads isa belief

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    12/32

    Axiomatic Approach to Probability:

    In the axiomatic approach, probabilities are numbers

    in the range [0, 1] Certain probabilities are assumed to be given (we

    dont ask how!)

    allows the calculations of other probabilities in amathematically & logically consistent manner

    It is probability calculas

    allows the computation of probabilities withoutrequiring philosophical discussions about the meaningof probability

    Consistent with all the approaches described above

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    13/32

    So , t h e P r o b a b i l i t y c a n b e d e f i n e d

    a s t h e M e a s u r e o f t h e p o s s i b i l i t i e s o f

    o c cu r r e n c e o f a n e v e n t i n a

    r a n d o m e x p e r im e n t .

    P r o b a b i l i t y o f a n e v e n t A i s

    d e n o t e d b y P ( A ) .

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    14/32

    Probability in Engineering:

    Thermal noise in electrical circuits

    Information Theory

    Communication systems design

    Noise

    Games of Chance

    Reliability of systems

    Failure probabilities

    Failure Rates

    Mean time to failure

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    15/32

    Networks & Systems Problems

    Random arrivals of packets/jobs

    Random lengths/service times

    Random requests for resources

    Probability of buffer or queue overflow

    Transmission or service delays

    Scheduling problem, priorities, QOS

    Flow control and routing

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    16/32

    Trial ?

    Experiement ?

    Outcome ?

    Sample space ? Event ?

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    17/32

    Experiments and Trials

    Fundamental notion: An experiment is

    performed and its outcome observed

    This is called a trail of experiment

    The experiment may be performed by a

    human agent, e.g., tossing a coin

    or rolling a dice

    The experimental outcome might just be

    the measurement of a naturally occurringrandom phenomenon, e.g. a noise voltage

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    18/32

    The Sample Space

    The set of all possible outcomes of anexperiment is called sample space of the experiment

    Examples: The experiment isTossing a coin: ={H,T}

    rolling a dice: ={1,2,3,4,5,6}

    noise voltage: ={x:-1x1}

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    19/32

    Exercise

    Example: The experiment is

    rolling a dice: ={1,2,3,4,5,6}

    suppose that each outcome is equally likely:

    P(1)= P(2)= P(3)= P(4)= P(5)= P(6)

    What is probability of rolling an evennumber?

    What is probability of rolling an prime

    number?

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    20/32

    An even number is said to have beenrolled if the outcome is any of {2,4,6}

    P (even number)=1/2; more explicitly

    P (even number)=3/6 since 3 of the 6outcomes are in the subset {2,4,6}

    An prime number is said to have been

    rolled if the outcome is any of {2,3,5} P (prime number)=1/2 also

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    21/32

    Event

    A subset of is called an event

    Example: A={2,4,6} & B={2,3,5}are said to be events defined onsample space ={1,2,3,4,5,6}

    events defined on the sample spaceis merely a probabilists way of sayingsubsets of the sample space

    Ac={1,3,5} & Bc={1,4,6} also areevents define on

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    22/32

    When does an event occur?

    An event A is said to have occurred on a trial

    if the outcome of the trial is a member of thesubset A

    Event A occurs if the observed outcome is

    some member of A; we dont care whichmember of A it is

    If the observed outcome is not a member ofA, then we say A did not occur, orequivalently, we say that Ac occurred

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    23/32

    Outcomes vs. Events

    Every trial results in only one outcome, i.e.,

    only one of the elements in can be observedoutcome

    The observed outcome is a member of several

    different subsets, i.e., events & all theseevents are said to have occurred

    Fundamental notion: on each trial of the

    experiment, one outcomes occurs, butmany events occur

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    24/32

    Example: If outcome of rolling adice is 4, then

    Events A ={2,4,6} & Bc={1,4,6} both

    have occurred

    Events Ac={1,3,5} & B={2,3,5} did

    not occur Event A U Bc = {1,2,4,6} has

    occurred

    Event A and Bc = {4,6} has occurred

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    25/32

    Two special events

    can be regarded as a subset of

    On any trial, the event always occurs The event is called the certain event or the

    sure event

    , the empty set, is also a subset of On any trial, the event never occurs

    The event is called null event or the impossibleevent

    A sample space of n elements has 2n

    different subsets including &

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    26/32

    Probabilities of the special events

    always occurs; c= never occurs

    Conclusion: the probabilities assignedto & should be 1 & 0respectively, regardless of how wechoose to assign probabilities to theoutcomes

    P()=1 will be used as an axiom inthe axiomatic approach to probability

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    27/32

    Arbitrary probability assignment

    Nonclassical approach: The n outcomes

    have probabilities p1, p2 ...pn where pi0& pi =1

    The probability of an event A is the sum

    of the probabilities of all the outcomesthat comprise A

    P(A)=sum of the pi for all members of A

    Example: A={x2,x4,x22}P(A)=p2+p4+p22

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    28/32

    Disjoint Events

    Events A & B are said to be disjoint or

    mutually exclusive if A & B have no element in common

    A ={1,3,5} & B={2,4,6} are disjoint events

    A U B = {1,2,3,4,5,6}

    P(A U B ) =p1+p2+p3+p4+p5+p6= P(A)+P(B)

    =I BA

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    29/32

    Probability Axioms for finite spaces

    Probabilities are numbers assigned to events

    that satisfy the following rules: Axiom I: P(A)0 for all events A

    Axiom II: P()=1

    Axiom III: If events A & B are disjoint, thenP(A U B ) = P(A)+P(B)

    Consequences: P()=0

    P(Ac)=1 - P(A); P(A)=1 - P(Ac)

    0P(A)1 for all events A

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    30/32

    Countably infinite sample spaces

    Let = {x1,x2,.xn,.} be the Countably

    infinite sample spaces P{xn}=pn where pn0

    For a finite subset A of , P(A) is just the

    sum of the probabilities of the outcomescomprising the event A, as before

    It seems reasonable to have this idea work

    for an infinite subset of A as well But we need a new improved axiom

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    31/32

    New improved Axiom III

    Let A1,A2,..An denote a countable sequence

    of disjoint events, i.e.,for all ij. Then, P(A1UA2U.. UAnU. )

    =P(A1)+ P(A1)+ P(A1)+..P(An)+.

    The new axiom implies that P() =0 &

    P(A U B ) = P(A)+P(B) for AB =

    =I ji AA

  • 7/24/2019 Unit-1_ Lec 1_ ITA

    32/32

    The Probability Space:

    Formal statement of the axiomatic

    theory A probability space (,F,P) consists of

    The sample space consisting of all

    possible outcomes of the experiment The -field of events F which includes

    all the interesting subsets of

    The probability measure P() thatassigns probabilities to the events