Question

3 tid Assume that X1,..., X. V(X) = . Poisson(X). The following facts about the Poisson distribution may be useful: E(X) -

Answer 1

Salection: X. ... Xn iid Poisson (~) and F(xi) - Var(xi) - ^ *i-icon ca) Assuming n is toe can use the central limit theorem. CLT. We know if we have al Sn- ke 0 X- Xn are iid n as n -00, a is. renknocon Å 8c on large; because it satiefies all the criterion of variables {X}x=1 with finite sequence of independent and identical tondom mean and finite variance, i.e E(X«)< and voor(xx) = 62200 k then define Sn = 2 Xx and - nu N(0, 1) as no in 6 equivalently, n (8-1) oC N(0, 1) as no Sinee for the above poisson problem, and E(X2) = Vor (xx) =) < v kulin. therefore according to CLT, S - 1.E(X4) (od) as no In VouT(Xk) where - Sn = + 2Xk * vn (x-1) asy, NO,1) Now we define that the pivotal (Justified) - ) quantity sample X = (X1). ,xn) foom a population Ps governed by pašameter o, is chosen. * function h(X1) -- 850) is called pivotal of the distribution of h(X;o) doesn't depend Based on this definition; suggest b(x - XN; ) T(- -) pivot for confidence interval of 1 because, the asymptotie dist" of h(xos -- ,Xn; 1) is NO, 1) and independent of a (from (&). (b) Nowo, we have already quessed h(Xos - Xn5A) and its dist? asymptotic Now for 95% confidence interval if we use the above pivot; it will be problemdie to separate and . To resolve the problems we will use varianee stabilizing transformations to find another pivot using square-root tansformation Xi - Poisson (ə) ti=i(Un. large; (.xi - x) D N (0,2). Let gc.) be a fune. I gr.) exists at a; then by delta method, ► No, (*(*))) take, > (g(x)) ² = e, g'(a) => g (2)= 204tk we can as a . as 12 is D g(xi) - ga

put c-/ko Var (qcxi)) becomes constant g(1) - => g(xi) - Nxi ti (Sxi da) N(o, 1) = 42(2lxi - A) X2 as no as yo here is unknoton and estimated by: Way SL & + n Ya. 2 xí w the young piyot, it L=0.05, 1-8-0.95 the new and E(52) = 0 so, tender null, T- 4 (vxi - Gaya xn, wer* nopiret So, 100(1-«)% confidence interval for x is obtained ; h asys (de, in) ? [152-dol & Toge ] = 1- * Yan. -- and Tap - To.025 pivot is kr (X) = J/ - upper-o-025 point of N(0,1) JAT (+2JX; -7). (0,1). => PICT-To025) = * = (sâ + Topone5 .) ²] with confidence eoefficient 95% (Ans.) (e) Interpretation : the above given interval will contain the tree value of with confidence coefficient 95% as s is large. 216 = 0.95 280 C.I is : [(SA - To 025 ) ² (da + Co.025 22 X -