Standard text : Black ,blue

Definitions : red.

Theory : Purple .

Sections .' : Generating functions 0

,Sometimes just write a :( an )

Let a =

Can,

n 30 ) be real or complex numbers.

The ordinary generating function of a is Gag )=§an5 ,for SER st the sum

converges .

Remand : Write f " ' for the ith derivative of f.

Then Galo

)=i!ai,

so G"

contains"

the info.

in a ( if it is C. at 0 ).

Etample -1 Pop ( De Moivre ) : ( cos 0 + is in G)"

= ( coscnottisinln -0 ) ( Aside : expect basic familiarityprod : Let an

= ( ei0 ) '= ( cos 0 + i sin G)

"

. with complex analysisThen for 151<1

,But here i is just a

GAG ) = § [ skosotisino )]"

=symbol with the property

1- SC cos -0 + is in -0) that i 2=-1.

Now consider the function

[ 1 - s( cos Otis in -0 ) ]§sn( coslnoltisinlno )) = #

Coefficient of S° is cos ( at i sin ( o ) = I

Coeff.

of sn,

n 't is

cos ( no ) + isin ( no ) - ( cos -0 + is in -0 ) ( cos An-1) 01 + is in ( ( n . , ) -0 ) )=

cos ( n -0 ) - cos -0 coskn- I ) 0 ) + sin -0 sinkn . i ) -0) + i ( sin ( no ) - Sino cos(( ntlf ) - cos -0 sin Ku . I ) -0) )

0

"

by compoundable formulae

So # = I so §Sn( coslnoltisinlno )) =§ [ s( cos -0 + is in -0 ))"

so cos ( not isincno )=( cos Otis in 0 )"

.

0

�2�

Def : For sequences

a- Cai ) and b= Cbi ).

the convolution ofa and b

,denoted

a*b,

is the sequencec with

Cn = aobntaibn. ,

t - - it an bo.

Remark : If c=a*b then

Gccs ) = focus" = E ( Eaaibn . , ) s

" = Gals ) Gb ( s )

.

Aside : So GIs arelike Fouriertransforms : they convert convolution to multiplication

Example 2 : Pap : If (f) 2= ( 2L )

Proof Let ai= ( ni ) to for i > n ).

Then E ( ? ) I §na ; an ... = ( a * a) n .

�1�

ien

Next , Ga G) = E (1)si= ( tsjnicn

so Ga*a( s ) = Gals )2= I its )"

= §n ( Y ) si

so (a* a) nt ( 2L ).

�2�.

Now compare�1� and �2�

.O

Def : If X a nv . taking values in IN =E oh , ... },

and f) is the probability mass function of X, �3�

then write Gxls ) = Gfxcs ) = §o fxlilsi =

, ,§P( * ilsi

= IE ( s× ).

[ sometimes just write GCs ) )

Remarks : If X. Y indep , integer r.us and Z= XTY , then

fzcn) - P(z=n ) = €nP(X=i ,Y=n - i ) = .§f×li-fln . i ).

So Gz = Gx Gr.

Examples Let X,

Y be independent,

X ~ Poisson

A).

Yn Poi ( µ ),

Z=X+Y

Then 6×61 = ⇐ Thief si = et 's " '

,Gy ( s ) =

en's "

so Gz ( s ) = eH+M ( s ' ' 'so Z ~ Poisson ( Xtµ )

.

Ex=mpk4_ If X ~ Geon ( P ),

so IPCX = k ) =p . ( I - plk"

for Ksl,

then

GGI = E(s× ) = §g st pa - psk"

= Psl- s ( I - p )

Reminders : D For a real sequence a = Cai,

i > 1 ),

the radiomen of Ga is

�4�

R : -

sup ( r 30 : Elailri < a ).

[ If a is PMF then Ela ; FEA ; =L so R ? 1 ]

�2� Ga is Ca inside radius of convergence .

�3� If R > 0 then an =

# Gama ).

[ So if Ga=Gb then a=b ].

�4� Abel 's Thm If ai 30 for all i and R > I then tiny Gals ) = §o9i .

theorem 5.1=18 If X hasgenerating function GG ) then :p R .

(a) IECXKG 'Cn,

d) more generally .IE#D....oCx-k+Dj=ga.y , , .

) "Gm4¥¥fIG"%).

*)kProof For s< I

, G" it

G) = §sink . ( i ) , ,

. PCX - i )

= IE"fs←k . ( × ) ,, ] ← since ( X) , ,=o if Xsk -1.

The radius of convergenceof Gal is 71 because this IE is finite for se l

So by

Abetsthm_yGatch = tiny Gma ) = If,

cnn.PH = i ) = El ( Xh. ]

. o

Corollary :Var ( X ) = IECXY - lE[ xp = IECXCX - i ) ] + IECX) - IECX ]2

= Gill ) +G'

( 1 ) - G'( 1)2

.

�5�

GFS and sums

r9

Exempt : Let Xi,

is . 1 be HD Be - ( p ), so

G×itH=lE[t×T=

ftp.pt .

With S=§nXi ,

then

Gslt) =

#

Gxittl=(q+pH'

.

Theorem 5.1.25 ( GF of RandySund ).

If ( Xi,

is 1) "

D.distributed as X

,and Nisan IN - valued v.v . indep .

of the Xi thenThe GF of S= Xi ' ' ' . + Xu is

Gs (f) = Gn (

Gxltl)

Proof : Gs CH = E[ ts ] = IE [ IEHSIN] ] =

E lE[tslN=n]P(N=n )

= § E[ tx 't ' ' ' th ]P(N=n )

=

€ EH×J . - - - IEFHYP ( N=n )=

EGHYPCN

=n ) = Gv ( Gxlt ) )

on>

, o"

Corollary :

"

IE [5) =lE[ X ) . IECN ].

proof : IES = Gs'

( i ) = Gj ( G. (D) GI Cn) = Gial Gill )o

"

Poisson thinning"

.

�6�

Exempt✓

If N ~ Poisson (7) and Hi,

i 311 n D Be - ( p ),

5= At - ' ' + XvThen

Gn ( x ) =e?⇐ "

Gslt ) = Gnlqtpt ) =

exp ( a ( cq+pt ) - i ) ) =

exp ( hp It - i ) ).

So S is Poi ( Xp ).

1- ptpt - I =p ( f . , )

Def ( Joint OGF ) If X ,Y are IN - valued r.us we define

G # Is .tl = lE[sxt " ].

Thus : X and Y are independent < ⇒ Get =G×Gy.

Proof : ( ⇒ ) If X ,Y ind.

then s× and t ' '

indep . so It [ sxt" ) =E[s×JtE[fY=G×ls)GtA

.

( ⇐ ) Exercise,

09.

[ sits'

) @ G. f) = PCX .

- i. Y=j ) [ sits ](G×( sight ) ) =lP(X=i)P(Y=j ).

If(these are equal then X. Y indep . 0

Useful notation : means" coefficient of siti in Gxor . Likewise [ SYGG ) is coeff

. of sn

.