featured - mcmaster university

featuredmore on momentgenerating factorsRecall that for a random variable X the

momatgenerating function of X is defined to be

MxCtl E IetThe Mgf has the property that for any h I

MY o FIX the nth moment

Ex Let be an exponential RV with parameters

ThenM Ct E Ce

tx

foe ne dx

a fo ea

ax

e F IF

MY G nation EGF

E Let Z Nfo 1 Then

Mz E E etZ

p e dcomplete the square

p f eti

dy xEztx ex E E

try e etkdxet't go e

Ex thdux

Now fth e is the dusty functionofT

Nlt o so Iguchi I

Hemel Mz t et

Now let X N u o Then X at OZ

Hence

My G Efe EfettutozyEfetnetozEn EEZ

I theCto

city t let

Mike e

PRI Let X y be Mdgs Then

Mxty t M t My t

Pfi Misty E E et City

EtetY

Efe E et'TMdt My t

Rinks If M H exists and B fontem some

region around t o then Mx Ct uniquelydetermines the distributionof X

Theoremlunquerent Suppose there B S o s t

Mx Ct My t CA ft E C S S

then Fx E FyCt htt C R

The proof is beyond the scope of this causes se

but we encourage you to read it on your own

see inversion formula for the Fourier transform

jFor any n random variables X Xu we

define the joint moment generating functionas

M t th E et

ti the Rh

Note that we can recover the momentgenerating

functions of each of the Xi and hence

the distributionsof the Xi's as

Mx Lt E et i M o o O t o o o

E ith spot

As in the single variable case we can

completely recover the joint distribution

of Xi Xz Xn from the joint mgf Though

the proof is too advanced for his cause

From this

Facti X Xn are independent iffMLt tn Mx Ct M

nth

2Somernequalitus

Proplmarkov'sIneqealthSuppose X Ba non negate

RV Then foray a 0

Paz a e Ektaff Let a o Set I

it t aO 01W

Then I Ba RV and

I E XT since if X a then 31and I I

So ECI I FEELThis games the result Smee ELI RX a

y

using this we get

Prof Chernoff'sBoard Let X be a RV Then

p x a E eatMxLH ft 70

Pf The function x to e m monotonemareamyfor t 70 so I

PCX a P et x eta

FIFI MEETy

F If X N 0,1 then

Nazim

This also gives a loner bound for OICa

I e Z Ica