featured - mcmaster university
TRANSCRIPT
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