Generating Function STAT253/37 Winter 203 Lecture 8 Yibi Huang January 25, 203 Generating Function For a non-negative-integer-valued random variable T, the generating function of T i the expected value of T G() E[ T ] in which T i defined a 0 if T. P(T ), G() i a power erie converging abolutely for all < <. if T i finite w/ prob. G() P(T < ) < otherwie P(T ) G () (0)! Knowing G() Knowing P(T ) for all 0,, 2,... STAT253/37 203 Winter Lecture 8 - STAT253/37 203 Winter Lecture 8-2 More Propertie of Generating Function G() E[ T ] P(T ) E[T ] lim G () if it exit becaue G () d d E[T ] E[T T ] P(T ). E[T (T )] lim G () if it exit becaue G () E[T (T ) T 2 ] 2 2 ( )P(T ) If T and U are independent non-negative-integer-valued random variable, with generating function G T () and G U () repectively, then the generating function of T + U i G T +U () E[ T +U ] E[ T ]E[ U ] G T ()G U () STAT253/37 203 Winter Lecture 8-3 4.5.3 Random Wal w/ Reflective Boundary at 0 State Space 0,, 2,...} P 0, P i,i+ p, P i,i p q, for i, 2, 3... Only one cla, irreducible For i < j, define N ij minm > 0 : X m j X 0 i} time to reach tate j tarting in tate i Oberve that N 0n N 0 + N 2 +... + N n,n By the Marov property, N 0, N 2,..., N n,n are indep. Given X 0 i N i,i+ if X i + + N i,i + N i,i+ if X i where N i,i N i,i, N i,i+ N i,i+, and N i,i, N i,i+ are indep. STAT253/37 203 Winter Lecture 8-4 ()
Generating Function of N i,i+ Let G i () be the generating function of N i,i+. From (??), and by the independence of Ni,i and Ni,i+, we get that G i () p + qe[ +N i,i +N i,i+] p + qg i ()G i () G i () p qg i () Since N 0 i alway, we have G 0 (). Uing the iterating relation (??), we can find p G () qg 0 () p q 2 p (q 2 ) P(N 2 n) (2) pq 2+ pq if n 2 + for 0,, 2... 0 if n i even STAT253/37 203 Winter Lecture 8-5 Similarly, G 2 () p qg () p( q2 ) q( + p) 2 p q( + p) 2 + pq 3 q( + p) 2 p (q( + p) 2 ) + pq 3 (q( + p) 2 ) pq ( + p) 2+ + p + pq + ( + p) 2+3 pq [( + p) + ( + p) ] 2+ p if n P(N 23 n) pq [(+p) + (+p) ] if n 2 + for, 2,... 0 if n i even STAT253/37 203 Winter Lecture 8-6 Mean of N i,i+ Recall that G i () E(N i,i+). Let m i E(N i,i+ ) G i (). i () p( qg i ()) + p(qg i () + qg i ()) ( qg i ()) 2 G p + pq2 G i () ( qg i ()) 2 Since N i,i+ <, G i () for all i 0,,..., n. We have m i G i () p + pqg i () ( q) 2 + qg p i () p + q p ( p + q p m i 2) [ + qp p + (qp )2 +... + ( qp ] )i + ( q p )i m 0 Since N 0, which implie m 0. (q/p) i m i p q + ( q p )i if p 0.5 2i + if p 0.5 p + q p m i STAT253/37 203 Winter Lecture 8-7 Mean of N 0,n Recall that N 0n N 0 + N 2 +... + N n,n When E[N 0n ] m 0 + m +... + m n n p q p > 0.5 E[N 0n ] n p q 2pq [ ( q (p q) 2 p )n ] if p 0.5 n 2 if p 0.5 2pq (p q) 2 linear in n p 0.5 E[N 0n ] n 2 quadratic in n p < 0.5 E[N 0n ] O( 2pq (p q) 2 ( q p )n ) exponential in n STAT253/37 203 Winter Lecture 8-8
Example: Symmetric Random Wal on (, ) State pace..., 2,, 0,, 2,...} P i,i+ P i,i /2, for all integer i clae, recurrent, null-recurrent or poitive-recurrent? For i, j, define N ij minm > 0 : X m j X 0 i}. Note N 00 + N 0 Given X 0 N 0 if X 0 + N 2 + N 0 if X 2 Note N 2 and N 0 are independent and have the ame ditribution a N 0 (Why?) STAT253/37 203 Winter Lecture 8-9 (3) Generating Function of N 0 Let G() be the generating function of N 0. From (??), we now that G() 2 + 2 E[+N 2+N 0 ] 2 + 2 G()2 which i a quadratic equation in G(). The two root are G() ± 2 Since G() mut lie between 0 and when 0 < <. Note that G () G() 2 2 + 2, which implie E[N 00 ] + E[N 0 ]. Symmetric random wal i null recurrent. E[N 0] lim G () STAT253/37 203 Winter Lecture 8-0 The power erie expanion of G() 2 can be found via Newton binomial formula ( ) α ( 2 ) α ( 2 ) where ( ( α 0) and for, α ( ) /2 ) i0 (α i)/!. 2 ( 2 )( 2 2)( 2 3)... ( 2 + )! ( ) 3 5... (2 3) 2! ( ) 3 5... (2 3)(2 ) 2!(2 ) ( ) (2 )! 2 ( )! 2!(2 ) ( ) 2 2 (2 ) ( ) 2 From all the above we have G() ( /2 ) ( 2 ) 2 2 (2 ) ( ) /2 ( ) ( ) 2 2 the ditribution of N 0 i ( ) 2 P(N 0 2 ) 2 2 (2 ) for 0,, 2,... P(N 0 2) 0 2 STAT253/37 203 Winter Lecture 8 - STAT253/37 203 Winter Lecture 8-2
4.7 Branching Procee Recall a Branching Proce i a population of individual in which all individual have the ame lifepan, and each individual will produce a random number of offpring at the end of it life Let X n ize of the nth generation, n 0,, 2,.... Let Z n,i # of offpring produced by the ith individual in the nth generation. Then X n+ X n i Z n,i (4) Suppoe Z n,i are i.i.d with probability ma function P(Z n,i j) P j, j 0. We uppoe the non-trivial cae that P j < for all j 0. X n } i a Marov chain with tate pace 0,, 2,...}. Mean of a Branching Proce Let µ E[Z n,i ] j0 jp j. Since X n X n i Z n,i, we have [ Xn ] E[X n X n ] E Z n,i X n E[Z n,i ] X n µ i If X 0, then E[X n ] E[E[X n X n ]] E[X n µ] µe[x n ] E[X n ] µe[x n ] µ 2 E[X n 2 ]... µ n E[X 0 ] µ n If µ < E[X n ] 0 a n lim n P(X n ) 0 the branching procee will eventually die out. What if µ or µ >? STAT253/37 203 Winter Lecture 8-3 STAT253/37 203 Winter Lecture 8-4 Variance of a Branching Proce Let σ 2 Var(Z n,i ). Again from the fact that X n X n i Z n,i, we have and hence E[X n X n ] X n µ, Var(X n X n ) X n σ 2 Var(E[X n X n ]) Var(X n µ) µ 2 Var(X n ) E[Var(X n X n )] σ 2 E[X n ] σ 2 µ n. Var(X n ) E[Var(X n X n )] + Var(E[X n X n ]) σ 2 µ n + µ 2 Var(X n ). σ 2 (µ n + µ n +... + µ 2n 2 ) + µ 2n Var(X 0 ) ( ) σ µ 2n 2 µ n µ n µ if µ Var(X 0 ) + nσ 2 if µ STAT253/37 203 Winter Lecture 8-5 Generating Function of the Branching Procee Let g() E[ Z n,i ] P be the generating function of Z n,i, and G n () be the generating function of X n, n 0,, 2,.... Then G n ()} atifie the following two iterative equation (i) G n+ () G n (g()) for n 0,, 2,... (ii) G n+ () g(g n ()) if X 0, for n 0,, 2,... Proof of (i). [ E[ X n+ X n ] E ] [ Xn i Z Xn ] n,i E i Z n,i X n i E[Z n,i ] (by indep. of Z n,i ) X n i g() g()x n From which, we have G n+ () E[ X n+ ] E[E[ X n+ X n ]] E[g() X n ] G n (g()) ince G n () E[ X n ]. Proof of (ii): HW today STAT253/37 203 Winter Lecture 8-6
Example Suppoe X 0, and P 0 /4, P /2, P 2 /4. Find the ditribution of X 2. l. g() 4 0 + 2 + 4 2 ( + ) 2 /4. Since X 0, G 0 () E[ X 0 ] E[ ]. From (i) we have G () G 0 (g()) g() ( + ) 2 /4 G 2 () G (g()) 4 ( + 4 ( + )2 ) 2 (5 + 2 + 2 ) 2 (25 + 20 + 42 + 4 2 + 4 ) P(X 2 ) The coefficient of in the polynomial of G 2 () i the chance that X 2. 0 2 3 4 25 P(X 2 ) and P(X 2 ) 0 for 5. STAT253/37 203 Winter Lecture 8-7 20 4 4 Extinction Probability Let π 0 lim n P(X n 0 X 0 ) P(the population will eventually die out X 0 ) Propoition The extinction probability π 0 i a root of of the equation g() (5) where g() P i the generating function of Z n,i. Proof. π 0 P(population die out) j0 P(population die out X j)p j j0 πj 0 P j g(π 0 ) STAT253/37 203 Winter Lecture 8-8 Propoition I Let µ E[Z n,i ] j0 jp j. If µ, the extinction probability π 0 i unle P. Proof. Let h() g(). Since g(), g () µ, h() g() 0, ( h () jp j j ) ( ) jp j µ for 0 < j j Thu µ h () 0 for 0 < h() i non-increaing in [0, ) h() > h() 0 for 0 < g() > for 0 < There i no root in [0,). STAT253/37 203 Winter Lecture 8-9 Propoition II If µ >, there i a unique root of the equation g() in the domain [0, ), and that i the extinction probability. Proof. Let h() g(). Oberve that h(0) g(0) P 0 > 0 h (0) g (0) P < 0 Then µ > h () µ > 0 h() i increaing near h( δ) < h() 0 for δ > 0 mall enough Since h() i continuou in [0, ), there mut be a root to h(). The root i unique ince h () g () h() i convex in [0,). j2 j(j )P j j 2 0 for 0 < STAT253/37 203 Winter Lecture 8-20