Coupon Conspiracy

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    Introduction

    Today we are going explain twocoupon collecting examples thatincorporate important statisticalelements. We will first begin by usinga well-known example of a couponcollecting-type of game.

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    We are going to use theMcDonalds Monopoly Best

    Chance Game to connect a realworld example with our Coupon

    Collecting Examples.

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    McDonalds ScamList of Prizes Allegedly Won

    By Fraudulent Means

    1996Dodge Viper(1) $100,000 Prize(2) $1,000,000 Prizes

    1997 (1) $1,000,000 Prize

    1998 $200,0001999 (3) $1,000,000 Prizes

    2000 (3) $1,000,000 Prizes

    2001 (3) $1,000,000 Prizes

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    The Odds of Winning

    Instant Win 1 in 18,197 Collect & Win 1 in 2,475,041

    Best Buy Gift Cards 1 in 55,000

    McDonalds Monopoly

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    Review

    Sample space the set of all possibleoutcomes of a random experiment (S)

    Event any subset of the sample space Probability is a set function defined on

    the power set of SSo if ,then P(A) = probability of A

    S A

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    Review ContinuedBernoulli Random Variable Toggles between 0 and 1 values Where 1 represents a success and 0 a failure Success probability = p Probability of Failure = (1-p)Bernoulli Trials Experiments having two possible outcomes

    Independent sequences of BernoulliRVs with the same success probability

    E[X] = p Var(X) = p(1-p)

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    Review Continued

    Geometric Random VariablesPerforming Bernoulli Trials:P=success probabilityX=# of trials until 1 st success

    Assume 10 P

    2

    1

    1

    p p

    X Var

    P X E

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    Expectations of Sums of Random

    Variables For single valued R.V.s:

    Suppose ( i.e. g(x,y) = g )

    For multiple R.V.s:

    R R :g 2

    y}Yx,P{Xy)g(x,Y)]E[g(X,x y

    i ii }xP{Xx]E[ X

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    Properties of Sums of RandomVariables

    Expectations of Random Variables can besummed.

    Recall:

    Corollary:

    More generally:

    E[Y]E[X]Y]E[X y}Yx,P{Xy)g(x,Y)]E[g(X, x y

    n

    1ii

    n

    1ii ]E[X]XE[

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    Sums of Random Variables

    - Sums of Random Variables can be summedup and kept in its own Random Variable

    i.e.

    Where Y is a R.V. and is the instanceof the Random Variable X

    0iXY

    i

    iXthi

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    Variance & Covariance of

    Random Variables Variance a measure of spread and variability

    Facts: i)

    ii) Covariance

    Measure of association between two r.v.s

    ] E[X]) E[(X (X) 2var

    )(E[X] ] E[X (X) 22var Var(X)ab)(aX

    2

    var

    )(Y-E[Y]) E[(X-E[X](X,Y)cov

    X,Y Y X Y X cov2var var var

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    Properties of Covariance

    Where X, Y, Z are random variables, C constant:

    (X)(X,X) var cov

    (X,Z)(X,Y) Z)(X,Y covcovcov (X,Y)c *(cX,Y) covcov

    (Y,X (X,Y) covcov [X]E[Y E[XY] E (X,Y)cov

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    McDonalds Coupon Conspiracy I

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    Suppose there are M different types of coupons, eachequally likely.

    Let X = number of coupons one needs to collect in order

    to get the entire set of coupons.Problem:

    Find the expected value and variance of X

    i.e E [X]

    Var (X)

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    Solution:

    Idea: Break X up into a sum of simpler random variables

    Let X i = number of coupons needed after i distinct types have been

    collected until a new type has been obtained.

    Note:1

    1

    m

    ii X X

    the X i are independent so

    1

    1

    1

    1

    m

    ii

    m

    ii

    X

    X

    Var X Var

    E X E

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    Observe:

    If we already have i distinct types of coupons, then using GeometricRandom Vari ables ,

    P(next is new)m

    im )(

    Regarding each coupon selection as a trial

    X i = number of trials until the next success

    and

    P(next is new)m

    im )(

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    We know that the

    )()(

    22

    2

    2

    1

    1

    imimm

    p X

    X

    mimi p

    Var

    imm

    p E

    i

    i

    so

    1

    02

    1

    0

    1

    0

    1

    0

    m

    i

    m

    ii

    m

    i

    m

    i

    i

    im

    X

    X

    miVar X Var

    im

    m E X E

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    so

    1

    0

    1

    0

    1m

    i

    m

    i imm

    im

    m X E

    m

    i im

    mm

    mmmm

    1

    1

    1..........

    31

    21

    1

    above,expressionthereversing

    11

    21

    ..........2

    11

    11

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    Eulers Constant

    mmi

    m X E

    mi

    so

    mi

    m

    i

    m

    i

    m

    im

    log1

    log1

    log1

    1

    1

    1lim

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    1

    12

    1

    02

    ,

    m

    i

    m

    i

    im

    im

    mi

    mi X Var

    So

    slidenextin thetrickausewillwehis,simplify tto

    1

    12

    m

    i imim

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    imim

    im

    imim

    imm

    imm

    mmii

    22

    2

    22

    sum.theof numerator

    thefrommgsubtractinandtomadding:Trick

    imimimmi 122

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    1

    1

    1

    12

    2

    1

    1

    1

    12

    11

    1

    m

    i

    m

    i

    m

    i

    m

    i

    imimm

    imim

    m

    mmm

    Applying the trick from the previous slide to,

    1

    12

    m

    i imim we get,

    pulling out the m in the first sum,

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    Expanding the sums from the previous slide

    1211211.....111.....11 222

    2

    mmm

    mmm

    1211211

    ......111

    ......11

    above,sumtheof orderthereversing

    222

    2

    mm

    mm

    1

    1

    1

    12

    2 11 m

    i

    m

    i im

    im By the Basel series we will

    simplify this in the next slide.

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    Explaining the Basel SeriesBasel Series

    mm mm X Var 2

    22

    2

    2

    222

    )log(6

    6 toconvergesSeriesBaselthem,largeafor

    6

    ......111

    321

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    In ConclusionSo clearly the variance is:

    and the expected value is:

    These approximations are dependent on the fact thatm is a large number approaching infinity. Where mis the number of different types of coupons.

    m X Var 2

    mm X E log

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    McDonalds Coupon Conspiracy II

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    The 2 nd Coupon Conspiracy

    Given:Sample of n couponsm possible typesX := number of distinct types of coupons

    Find:E[X] (expected value of x)Var(X) (variance of x)

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    Redefining X

    Decompose X into a sum of Bernoulli indicators1 if a type is present

    0 otherwise

    1

    i

    i Let X

    i m

    1

    Note:m

    ii

    X X

    1 2

    (knowing one coupon type occured lowers

    the opportunity for the other coupons types to occur)

    Remark: , , , are dependentm X X X

    1

    Note:m

    ii

    E X E X

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    Redefining Var(X)

    1 1 1

    Re : ,

    ,m m m

    i i ii i i

    call Var X Cov X X

    Var X Var X Cov X X

    1 1

    1 1

    ,

    ,

    m m

    i ji j

    m m

    i ji j

    Cov X X

    Cov X X

    (by covariance bilineararity)

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    Summary thus far

    1 1

    ,m m

    i ji j

    Var X Cov X X

    1

    m

    ii

    E X E X

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    Success Probability

    Recall: is Bernoulli, with success Probabilitly Pi X

    { 1} 1 { 0}i i P P X P X

    m-1 [ ] 1

    mi

    nThus E X

    th Note: { 0} {type does not occur on j trial}i P X P i

    1 1

    1 { 0}

    m-11m

    n n

    i j j

    P P X

    n

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    Recapm-1

    [ ] 1 mi

    n E X

    m-1 m-1[ ] 1

    m mi

    n nVar X

    1

    1

    , [ ] [ ]

    m-11 m

    m-11

    m

    m

    i

    i

    m

    i

    So E X E X

    n

    nm

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    Var(X)

    To calculate ( ), we need to find , i jVar X Cov X X , [ ] [ ] [ ]i j i j i jCov X X E X X E X E X

    In particular when i j

    [ ]E X X

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    k

    i j i j

    A event that type is present

    [ ] { 1}

    A A 1 A A

    i j i j

    Let k

    E X X P X X c

    P P

    i j i jBy De Morgan, A A A Ac c c

    1 A Ai jc c

    Using the Inclusion/Exclusion rule: P A B P A P B P A B

    [ ]i j E X X

    1 A A A Ai j i j

    c c c c P P P

    [ ]

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    [ ] cont.i j E X X

    [ ] 1 A A A Ai j i j1 1 2

    1

    1 21 2

    i jc c c c E X X P P P

    n n nm m m

    m m m

    n nm mm m

    i j

    i j

    1 1Re : A , A

    2A A

    c c

    c c

    n nm mcall P P

    m mnm

    P m

    ( )

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    ( )i jCov X X 1

    Re : [ ] [ ] 1

    1 2[ ] 1 2

    i j

    i j

    nmcall E X E X

    mn nm m

    E X X m m

    211 2

    ( , ) = 1 2 1

    21 2 1 1 1 2 1 2

    22 1

    i j

    n n

    n

    nn n mm mCov X X

    m m m

    n nm m m mm m m m

    nm m

    m m

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    Re : Var ( , )

    i i jcall X Cov X X

    1 1

    ( )= ( , )m m

    i ji j

    Var X Cov X X

    Back to the Var(X)

    1 1

    1 2 2 2

    1 3 2 3 3 3

    1 4 2 4 3 4 4 4

    1 2 3 4

    ( , ),

    ( , ), ( , ),

    ( , ), ( , ), ( , ),

    ( , ), ( , ), ( , ), ( , ),

    ( , ), ( , ), ( , ), ( , ), ( , )m m m m m m

    Cov X X

    Cov X X Cov X X

    Cov X X Cov X X Cov X X

    Cov X X Cov X X Cov X X Cov X X

    Cov X X Cov X X Cov X X Cov X X Cov X X

    1 1

    ( , ) 2 ( , )m m m

    i i i ji i j i

    Cov X X Cov X X

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    Var(X) in terms of Covariance

    1 1

    ( , ) 2 1 ( , )m m

    i i i ji i

    Cov X X m m Cov X X

    1 1

    ( ) ( , ) 2 ( , )m m m

    i i i ji i j i

    Var X Cov X X Cov X X

    Simplifying the summation, We get:

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    221 1 2 1

    1 2 2

    ,

    21 2 12( ) ( 1)

    n

    n

    n n nm m m mm m m

    m m m m

    Simplified

    n nm m mVar X m m m m

    m m m

    Var(X) equals

    1 1Recall: ( ) = 1

    n

    i

    nm mVar X m

    m m

    1 1( ) ( , ) 2 1 ( , )m m

    i i i ji i

    Var X Cov X X m m Cov X X

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    21 2 12( ) ( 1)n n nm m m

    Var X m m m mm m m

    In Case you Forgot:

    m-1[ ] 1

    m

    n

    E X m

    [ ] log E X m m2( )Var X m

    Independent Case:

    Dependent Case:

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    Analytic vs. Simulation

    ApproachesFor a simulation we will consider a fair setof the McDonalds Coupons from Problem

    I.We will analytically find E[X] and Var(X)and compare these two values with the

    results from our computer simulation.

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    Analytic Approach

    Given M = 10, compute

    mm mm X Var 22

    2 )log(6

    mmim X E m

    i

    log11

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    References

    Sheldon Ross Author of Probability Models for Computer

    Science, Academic Press 2002 2-3 pages from C++ booksMcDonalds Dr. Deckelman

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    Any Questions???