Question

2. One common distribution that appears in branching process theory is a DRV with pmf: fx(x;j) = w. e-h (ux)2–1 — where x E {1, 2, ...} and pi € [0, 1] a. Find the MLE for u given iid X1, ..., Xn. Then, find the MLE for the particular data X1 = 2, X2 = 1, X3 = 6.

Answer 1

Answers Given, hy (x,c) = e they (uz)" xE {1,...y 2! Here ue [o,i] a) Likelihood function can be wozitten as 4*149*-xmla) - 347X: 1*2*, -UIx, Iz- ?-1 high-24 4 CEx: + n) kogut hog (,3 *-) X 2 + 0 - 0 Beyle - *:* (**) 40-0 for came togt zo -EX: + (Exion) : 0 For Camle 20 e an i adi u = al I lion Exi Scanned with CamScanner

1 Come si - ا = ما mi 8,72, 2231, 2326 Ex:= 2+176-9 121 n=3 îmle = 1-3 = 2/ lêmle = 0.66 1(21188 --281m) - é “TX L27: -" o " D A reol Put 2, 2, 82=1, &z=6, n=3 9-3 6-1 -M9 е у 2-1 2 x 1 -1 x ц ( Q! x!! X6 ! -qu 6 L = ex al x 54 La SH ude94 csscanned with CamScanner

0.00/5 t 0.004 0.0037 I 0.0012 + amle 1 CScanned with CamScanner I 2 3