Good Conditional Expectation Notes

8/13/2019 Good Conditional Expectation Notes

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Stochastic Processes

(Lecture #4)

Davy Paindaveine

Universite Libre de Bruxelles

Stochastic Processes (Lecture #4) p. 1/


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Today

Today, our goal will be

to finish with conditional expectations,

to have a quick look at conditional variances,

and

to discuss limits of sequences of r.v.s, and

to study famous limiting results.



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Conditional expectation

Conditional expectationallows for exploiting some information(e.g., the information that some event occurred) in order to

improve the (unconditional) best guessE[X].

This available information can take various forms:

(the occurrence of) an event.

(the value of )a r.v.

(the occurrence of some event in) a sigma-algebra.



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Conditional expectation (w.r.t. a r.v.)

LetXbe an integrable r.v. on(,A, P).LetY be adiscreter.v. on(,A, P), say with distribution

distribution ofY

values y1 y2 . . .

probabilities p1 p2 . . .

Then we defineE[X|Y]as the r.v.

E[X|Y] : R

E[X|Y =Y()].The last conditional expectation is w.r.t. an event, and hence is

well defined.



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LetXbe an integrable r.v. on(,A, P).LetYbe anabsolutely continuousr.v. on(,A, P), say with pdff.

Then we defineE[X

|Y]as the r.v. such that

(i)E[X|Y]is(Y)-measurable.(ii)

A

E[X|Y]() dP() = A

X() dP(), for allA (Y).

In practice:

IfX is discrete,E[X|Y =y] =i xiP[X=xi|Y =y].

IfXis absolutely continuous,

E[X|Y =y] = 1fY(y)

R

x f(X,Y)(x, y) dx=

R

x f(X|Y)(x, y) dx,

wheref(X

|Y)

(x, y)denotes the pdf ofXconditionally onY =y.



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Example: letX Unif(0, 1). IfX=x, you flipm times a coin suchthatP[Head] =x. LetNbe the number of heads.

E[N] =? Distribution ofN?

E[N|X=x] =mn=0

n P[N=n|X=x] =mn=0

n

m

n

xn(1 x)mn =

=

mn=1

m

m 1n 1

xn(1x)mn =mx

mn=1

m 1n 1

xn1(1x)(m1)(n1) =mx.

E[N|X] =mX ((X)-measurable).

Consequently, E[N] =E

E[N|X]

= E[mX] =m E[X] = m2.



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Distribution ofN?

P[N=n] = R

P[N=n

|X=x] fX(x) dx=

1

0

P[N=n

|X=x] dx

=

10

m

n

xn(1x)mn dx=

m

n

10

xn(1x)mn dx=. . .= 1m + 1

,

for alln= 0, 1, . . . , m.

Hence,Nis uniformly distributed on {0, 1, . . . , m}.(which now makes clear whyE[N] = m2).



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Conditional expectation

Conditional expectationallows for exploiting some information(e.g., the information that some event occurred) in order to

improve the (unconditional) best guessE[X].

This available information can take various forms:

(the occurrence of) an event.

(the value of ) a r.v.

(the occurrence of some event in)a sigma-algebra.



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Let F1 F2be two-algebras in A.Extra properties:

EE[X|F2

]F1= E[X

|F1].

E

E[X|F1]F2

= E[X|F1].

Conditional expectation w.r.t. to a-algebra also allows for

defining conditional expectationw.r.t. a collection of r.v.s.

More specifically, we define

E[X|Y1, . . . , Y n] := E[X|(Y1, . . . , Y n)],

where(Y1, . . . , Y n)is the smallest-algebra containing

{Y1i (B)

|B

B, i= 1, . . . , n

}.



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To understand how one can computeE[X|F]in practice,see exercise 6, in homework 1.



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Conditional variances

LetXandYbe r.v.s on(,A, P).As we have seen, one can exploit the information sitting inY

in order to improve the (unconditional) best guessE[X].

the improved guess is the conditional expectationE[X|Y].Similarly, onceYis observed, the knowledge of the

(unconditional) dispersion ofX, namely

Var[X] = E

(X E[X])2

,

can be improved into theconditional variance

Var[X|Y] = E

(X E[X|Y])2Y.



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Today

Today, our goal will be

to finish with conditional expectations,

to have a quick look at conditional variances,

and

to discuss limits of sequences of r.v.s, and

to study famous limiting results.



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Convergence of sequences of r.v.s

LetX1, X2, . . .bei.i.d.r.v.s, that is, r.v.s that areindependent andidenticallydistributed. AssumeX1 is square-integrable, and

write:= E[X1]and2 := Var[X1].

Let Xn := 1nn

i=1 Xi. Then

E[Xn] = 1nn

i=1E[Xi] = E[X1] =, and

Var[Xn] =

1n2 Var[

ni=1 Xi] =

1n2

ni=1Var[Xi] =

1nVar[X1] =

2

n,

which converges to0asn .

Consequently, we feel intuitively that Xn X, whereX=a.s.

How to make this convergence precise?



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Consider a sequence of r.v.s(Xn)and a r.v. X, defined on(,A, P).How to defineXn X(asn )?

Xna.s.

X(almost surely) P[{ |Xn() X()}] = 1.Xn

P X(in probability) For all >0,P[|Xn X| > ] 0.

Xn

Lr

X(inLr

,r >0) E[|Xn X|r

] 0.Xn

D Xin distribution (or in law) FXn(x) FX(x)for allxatwhichFX is continuous.



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Example:

LetY1, Y2, . . .be i.i.d. r.v.s, with common distribution

distribution ofYi

values 0 2

probabilities 12

1

2

DefineXn=ni=1 Yi.

The distribution ofXn is

distribution ofXn

values 0 2n

probabilities 1 12n

1

2n

We feel thatXn X, whereX= 0a.s... But in what sense?Stochastic Processes (Lecture #4) p. 19/


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We feel thatXn X, whereX= 0a.s...But in what sense?

In probability?

For all >0,P[|Xn X| > ] =P[Xn> ] P[Xn >0] = 12n 0,asn .

Xn P X, asn .

InL1?

E[

|Xn

X

|] = E[Xn] = 0

1

12n + 2n

12n = 1,

which does not go zero, asn . the convergence does not hold in theL1 sense.



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It can be shown that:

Xn a.s. X Xn P X Xn Lr

X

Xn D X


C f f


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Some arrows can sometimes be reverted:

Xn P X there exists a subsequence(Xnk)for whichXnk

a.s. X. Xn P Xand theXns are uniformly integrable Xn L

r X. Xn D a Xn P a(a is some constant).

Definition: theXns are uniformly integrable (i)supnE[|Xn|]< .(ii)P[A]

0

supn A |

Xn()

|d

0.

Recall: ifX is integrable,P[A] 0 A |X()| d 0(hence,only thesupnbrings some extra condition in (ii) above).


Li iti th


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Limiting theorems

Here are the two most famouslimiting resultsin probability and

statistics...


Li iti th


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Limiting theorems

Thelaw of large numbers(LLN):

LetX1, X2, . . .be i.i.d. integrable r.v.s.

Write:= E[X1].

Then

Xn := 1

n

n

i=1

Xia.s. .

Remarks:

Basically no assumption. Interpretation for favourable/fair/defavourable games of

chance.


Li iti th


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Limiting theorems

An example...

LetX1, X2, . . .be i.i.d., withXi Bern(p).Then= E[Xi] =p, so that

Xn := 1

n

ni=1

Xia.s. p.

Interpretation:

The empirical proportion of successes converges to the

theoretical proportion of successes (that is, the probability of

success).

Remark: this also shows that, if(Yn)is a sequence of r.v.s with

Yn Bin(n, p), thenYn/na.s.

p.Stochastic Processes (Lecture #4) p. 25/

Limiting theorems


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Limiting theorems

... and some applets:

http://hadm.sph.sc.edu/COURSES/J716/a01/stat.html

http://www.stat.berkeley.edu/stark/Java/Html/lln.htm ...


Limiting theorems


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Limiting theorems

Thecentral limit theorem(CLT):

LetX1, X2, . . .be i.i.d. square-integrable r.v.s.

Write:= E[X1]and2 := Var[X1].

ThenXn /

n

D Z, withZ N(0, 1).

Remarks:

Xn/n

=XnE[Xn]

Var[Xn].

It says sth about the speed of convergence in Xn a.s. . It allows for computingP[Xn B]for largen...

It is valid whatever the distribution of theXis!Stochastic Processes (Lecture #4) p. 27/


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Limiting theorems


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Limiting theorems

(B) Throw 9 dices and denote by Xthe sum of the results.Repetition of this yields i.i.d. r.v.sX1, X2, . . .

Note thatXi =Yi1+ . . . + Yi9, whereYij is the result of the jth

dice in theith roll. Then we have

= E[Xi] = E[Yi1] + . . . + E[Yi9] = 9 3.5 = 31.5

and

2 = Var[Xi] = Var[Yi1] + . . . + Var[Yi9] = 9 2.91666...= 26.25,

so that

n(Xn 31.5)

26.25

D Z, withZ N(0, 1).


Limiting theorems


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Limiting theorems

... and some applets:

http://www.math.csusb.edu/faculty/stanton/probstat/clt.html

http://www.stat.vt.edu/sundar/java/applets/(then go to Statistical Theory Central Limit Theorem

Main page).

http://www.jcu.edu/math/isep/Quincunx/Quincunx.html http://bcs.whfreeman.com/ips4e/cat 010/applets/

CLT-Binomial.html

...


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