Question

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

Answer 1

ANSWER:
Xos---- X9 id can 0 poction (1) and Ecxi) - varel xes-11-16? 32) f seuning in &s large, we use the central remit theoreem. because it satisfees all the exitencion of C17. We know if we sequence of independent and identical teanday finite yeen and finite and finite Variance. coer E(X4) sillies and vare LXq) = 02 coat Then define have a Nareeables with { xx}x=1 Sy= ²X, and KE q N(01) as n to Sn-nee no ore equivalently, metall) Y, NCoil) asor since for the above poossion problem, Nu Хт) are ied and Е (ХК) Vare LX 4) sX Lo the 101) There force according to CLT, Soj-NoE (X4) a YN (011) as no n van (AK) 0 EXK where, X= sing Sin ๆ Kel in (x-2) aly N (oil) as nas justifed ) Now we define what the pivotal quantity as -

seggest h in (T-d) To resolve the problem, We will cese vanedance star A sample X = (X0 --- 5xn) fou a population p. govannet by unknown parameter og de choren of function k (x1 - xp of eased pero tar iff the distreebution of A LA ;o) doesn't depend on O. can Based on thes de fonction, we C X, sro, Xn jA) pevot fore con fidence Enderval ofà. Because , the cegupto tee desto of n (Xen, Xojn) elle NCOUL) and codepend- ent of 1, (from ()). 302) Now, we have already guessed hlX , , . ... ) Xn ; A) asyiptotere dee totputoon fore 95% confidence interevay of we use the above påvot;. 2t will be pooblematie to separeate X and to to find another povot foane forencate on esong Xi ~podsson (AJ V I =1(1) as o de tange ; (Xerd) N. (02) Let g c.) be a fine .> glos exists at di delta Method gCxi l-g(0) and its Now bilizing treane formation, square recol Then by IN [0,169144))3]

8 is unknown and estimated by - take , & Egicas?? = ei g'(X) = e = g (A) = 2erłtk becomes cone tant . ing(1) = 11 3 g (xi) = TXT to alUX-6) DN(0ity) as n to gue (Para a xz os nyan Clothing to root you peet er 2.450 3) vare To Cai)] j mily Ź JXé E (A) = ist and 80, uydere meli, T-42 (897 - Vã] x ! So, 100 (1-0) of confrodence enterval fore & be obtained 1 y Nina, yo) P Train vyys 스 C Hal 51-9

8 =) = run h 2 xi-in) aN(1) C] = 1-9 weith confidence coefficient d = 0005 1-d = 0.95 and d = 10.025 2 Uper : 0.025 point of N(0,1) the new pivot is hit 2 32) & VAN 40. Then , ЭР l-na 219 arh - 0.95 el is 70.025 + To.025 am 2/ ๆ 95% (Ans 33) Interpretation : The above given Enterval will contain value of A with confidence coefficient 95% 스 T00025 + 7 P (A - Corons .) 22 x 1 the treee ces n Es large.