Question

(4) Consider the following random family tree: Let Y, denote the (random) number of people in the nth generation. Each person in the nth generation produces a random number of offspring, which has a Poisson(A) distribution. The total number of such children is then denoted Yn1 The number of offspring produced by any person is (statistically) independent of the number produced by another person. Moreover, Yo 1, that is, there is exactly one person in the zeroth generation. (a) Determine E(Y+1Yn ) and so also E(Y) (b) Relate the moment generating function of Yn+1 to that of Y (d) Give a formula for the variance of Yn+1 in terms of that of Y

Answer 1

(4) Mom ouestr on, nt 7m no.-of Madusid kynndiida Given fach praduses endom mumber f *X len (A), TEOKK nprkeny nti Xt2t.y 1uen Handom varialde (inkepentent) Paiim distubutin

X t X2t +Xx la)Elint in)Et E(XitX) n d Ih X uun (A), E(K) va(x) A ()EEaril)) - (n A) El)

E (n)EE( ) =NE (71) E (nti) 1 2 E(i) nti El70) E(1) nt1 1nt 1giren) И tenee Eln)

Iauen A) X tx = E ( e Aet A Cet1) tnti Inti 7 EYntl 2/3/3 tele niletI) Pewen (in4)

HCnee, Minti (t)M Ae-) e E var (n) Vai lElne)) + Var (7n) E cine E(n) Vau(in in)E) tvau) -AM n +vailin) n vau (n) nt E) -A voulin) nt1 2 vauln) 1

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