Question

Question 6 [15 marks] Let X1, X2,..., Xn be independent and identically distributed random vari- ables with common probability function ()p(1-p) m m-a ; x 0,1,. ., m otherwise 0 where m is known and p is unknown (a) Obtain the Sequential Probability Ratio Test of Ho p = po versus HA p P, where pi > po, with significance level 0.01 and power 0.95. Describe the test precisely; (b) For the case where po 3/8,pı = 1/2, m = 3,a = 0.01 and find approximations to the expected sample sizes. 0.95,

Answer 1

Solutions Criven that x,, x, Xn are th pdf fix,A)J) C1- p)"m:^; Ke0,1,-. iid -m for SPRT, have to find We Tf(xi,h) Am- Lih Loin T F(Ni,) m-2i Im DIC)- M-i Am= nm- h(1-M) h(1- oti 1-Pr - wu xi log (t/) nm- Ext) ly (1-h) Log Am +

(1-A) 1-P0 Then 1) Accept Ho if PgAm (Ros 1-a where .6 and B1-Powes 1-0.15 Hence kog Am o05 10.6 2.98 1-P if 2 i Lo711-1/1-to) (h-l.i-) 1-Pr S 2-92 + m Reject Ho if oAmLog 1-alsa if log Am S5 4.55 0 m

ii) Continue sampling if b Rog Ama Am L1.55 2.98 R) Expee ked Bample aigus given by eCn)= [1-powes) RoB eln)= (1-poN er) bt Powes a ecz) where e (z)2 (E) h)f(,) m-x Lo 1-P1 1-P m-2 iD XC1-9) 3-0 + 3-2 3-1 $3

Elz)-0-67( 1-p23-0.48 r(1-p] t06 p? ( 1- b) +0.26 p3 Henre Brb + (1-8) xa e(2) .95x-2.98 (I-o.95)x 4.5s ecn) 2.831 + 0 22 75 - 067 1-p-0.12(1-) 1.06 p ( 1- p), E3 78.9 t