MATH180B: Introduction to Stochastic Processes I

www.math.ucsd.edu/~ynemish/180b

This week:

HW1 due Friday, January 17, 23:59 pm

Hint for problem 5:

Today: Conditional distribution / random sums

Next: PK

-- t ta# A- Ei y:)

( find x and y)

conditionaldistribution-ldiscretec.ae )

Recall : for two events A ,Be I,conditional probabilités

of A given B is computed via

Déf.

Let X.Y be two discret r.v.is taking values in

{xr.az , _ . - I and { y . . yz .. . . } correspondingly .

The conditional

probability mass function of X given Y is defined by

By the law of total probability ({ Y=yj4Î , is a partition )

conditionaljointdistribution-Nota-tiniforr.is. × taKing values in { x. , xz . .

. - 3 and f- R-→ RO

the textbook uses notation Zflxi) = : Ifk) ,which mag cause

f- Iconfusion

Def ( joint coud .distribution )

(et X.Y ,Z be r.v.is.

Then the conditionat jointdistribution of ( X ,

Z) given 7- y is defined by

P x.zly (XiiZklyj ) : =

Renarde.

For any fixe d yj , any conditionat (joint)distribution (of X or ( X , Z ) or . . . ) given Y=yj isitself a ( joint) probabilités distribution .

Example- Let MEN

, piqt toi ')

(et N - BCM , q ) and det X - B ( Nip) .

In other wards ,

for ne 40,1 , - - i. Mb pan ( k tn ) = (f) p" ( t- p)"

what is the (marginal) distribution of X ? pxlk) - ?

By the law of total probabilité ,for kelo.li . . _ , M }

Conditionalexpectationcdiscretecasetef.

Let X. Y be discret r.v.is with values hais, tyj } .

Cet g : IR → IR be a function such that IE ( g (x)) la-

The conditionat exportation of g (X) given Hy; is defined

by

Simi larly for Elg ( X - Z ) IY --y ;) .Rent .

E ( gtx) IY )

By the law of total probabitity

PropertiesottheconditionalexpectatiletX.Y.Z-ber.v.isdefined on the same pro bability space .

[et g : R→ R and V : RIR be sit- Etg (X) 1) ca , ECIVCXM)1)a

Recall , for fixedyj , the (joint) distribution of X (or CX , -2 ))given Y -- y ; is a Cjoint ) probabilité distribution .Conditional expirations have the following properties :

(some analogons to the properties of usual exportation:)1. (Linear ity) E ( c. g. (X ) tczg.CZ ) IY = y;)

2. H gzo ,then Elg (x) I Kyi)

3. E (VCXMIIY - y ;)

Propertiesoftheconditionalexpectati-4.ECg (X) I Y = y;) = if X and Y are independent

5. Elg (X) htt) IY-- y;)6- E ( g (X) h (Y))

Proof-

:

Conditionnai (conf .)We can also define for a r - v . X and any event AeJpxlxlA) = P(¥sn ,

ECXIA) -_ Êxipxlxi IA )ExampleCExerci-e.tt)(et X- Pois (d) and Iet A = { Xisodd } .