Question

Problem 4 (10 Points) Derive the moment generating function and the characteristic function for a geometric random variable X with parameter p. Compute E(X) for p = 4/5. Prohlom 510 Pointe Tot Y Y Y be indarrandort randomiblog ebore V Tot

Answer 1

М. N solution Xu heometric secundom verliable with people fino a qup; x = 0,1,2 . ;q=1-9 Moment growding function te for Eetu gap Mylet 3 - Eletu) PE e Uzo P (Using Infinite formula) GP Juies 11-gect Charecteristic function na P (d) 2 Eleute) Ě eitse qp x=0 ne E 20-0 (qe it) il geit 1-qout

E(X) = d Mx (4) dt la o = d [ Ir gert 10 dt .pq (1-gest ? t-o [ne = 1 oto 니 Hence, for p 1 1-4 [ Arcs ) 4 sin