Download - 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???