MATH180B: Introduction to Stochastic Processes I

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

This week:

all office hours are on course web site

HW1 due Friday, January 17, 23:59 pm

Today: Probability review (cont.)

Next: PK 2.1, 2.3

: _

Joint normal distribution font-

Lost time :

Proposition 1 Lettre R'

,let Z c- IRZ" be positive définit ,-

"

t

let Y :(4. Yr ) be independent standard Gaussien r.v.is .Cet Ae R

"

be such that [ = AAT (always possible)Then A-Ytrî - Mpi , E) .

Corollaire .

Let X - Npi , Z) ,2- = AÂ - Then

A-'

(X - À) - MOI)

Corollaires . Any linear combi nation of Gaussien random

vector is again a Gaussien random vector.

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Corollaire 1f X -_ ¢ ) - Nlpî , E) , then

[ = ( Var (X ) Couch , Xz )

↳voix .) para) ) ← "VI. ¥Prof . Compute EKX -f) (X -f)

*

) = E ( Html (*t'Wta)

- (x, -µ) (X ,-

µ) Œuf)l' aimante"? """H:Let Y , Mz be Independent Mail) , dénote Y - t

,

and let A be such that I = AÀ .

Then by Prop ' , AY has the same distribution a:-X -F.there frere ,

E ( (X -F) (X -À) -_ ELAY (AMF ) -- ELAYYTÀ)=y AECYYT )A AI À = Es

check indxp .

Example[et Yi

,Ya be indep .

standard normal r.v.is

4in Moi 1)

Define X, =3 Y ,

- 2 Yz,Xz =3 Yzt 2. Y,

what is the distribution of ¥ ) ?Let Y > (%) - Mo ,

I).

Then we can write

X-p )=Ç)(%) - By Prop -t.

A

(¥) has bivariate normal distribution with mean ô - % )

and cou -

matrix E- AAT -- ( { f) /? } ) =L ? ;) .1h particulier , since Cov (X , ,X4=o ,

X,and XL are indep . Gaussien

(with variance B)

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Compute the density explicitiez : (et À -- (f) c- R' ,[ =L ?! %)» ,

Et * (¥ ) - Nlrîiz ) .

ftp.z#e+*- e-" Ï"

pû

aetz-oisi-oror-oioilt-f.is#--oioi(i-§ùË!Ë)Then , vdetz-o.rzh.PT

⇒ riff )

simiiarty.E-o.si#tI.iEi)---pr(ËÎË)'

km (( f- ap " "E- +EI ))↳ formula from the book ( 1.46 )

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Renois.

1) Definition 1, Proposition land its corollaries t - s

are not restricte d to random vector of dimension 2 :

we can general ize the definition by taking for any KemK KXK

À c- IR, 2- c- R ,

E > 0 , to define a random Gaussien

vector Xe Rr,X -NIF , E ) with the joint p . d. f.

£ (*) = [Ëz e- EH-ritz-tcx.rs

Proposition I and Corolla ries t - z remain unchanged .

(1) Multi variante ChT

Joint normal distribution font-

2) The joint normal distribution is unique ly identifiedby the mean § and the covariance matrix Z

.

Exe We have 3 r- v . X. Y ,-2

XM , 2- are all Mini ff-1,13j'ËEIË coin ¥ c.un -

- cou " "X , if Xe f-1,0)

z =p - (x+ Di if Xt to " ] but C ! ) (x.4) ¥ (X. Z )X , if XE f- ho )

3) 1f normal random variables are uncorrelated ( [ = I ) then

joint pd.f.is product of individual (marginal ) p.d.f.is ⇒ Independence

Estimatingtailprobabitities-Markovthm.LAX be a r . v.

, PCX >a) = I (XZO almostsurely ) .Then for any c > 0

cheby.sn#PCX ? c) ± ¥)

Ttsm.

Let X be a r.v.,ECX ) --µ ,

Var (x) = 6? Then

for any c> °

p ( × -µ > c) ± ÊImmediate corollaires

P ( X -µ > c) ± PCIX -ul x ) ± Éft

P ( X -µ ± -c) ± =

Momentgeneratingfunction-D-ef.tnet X be a ri.Then the function

Mx It ) -- E ( e" ) tt te R

is called the moment generating function of X ,

M - g. f. is used to characterize the distribution of a v.v . :

Thin.

Let X and Y be two r.v.is,let Mxlt) and My H )

be their m.g.f.is .

If there existes 5> o s -t .

il Mxlt) and My It) are finie for all te C- fr)

(ii ) Mx It ) - My Ctf for all te C- Sic ),

then X Y,

X and Y have the same distribution.

Computing moments from on _ g. f. ! if Mx It) is bounded

uround t - O , then,M'×(D) = ECX )

,

M (D) = ECX ' ) ,M (a) = EH

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