Question

Please solve this by explaining everything.

1 1. Suppose that X is a random variable with a Gamma-(k, 1) distribution where k > 0 is known, but > 0 is unknown Vx € (0,1), we have fx(x) 1k2k-1e-x T(k) Let us use 0 = 1/1 which is the standard approach, for example in Hogg and Tanis. Calculate the Fisher information I(0).

Answer 1

Note that the support for the Gamma distribution given in question is wrong. It should be x > 0 instead of x lies in (0, 1).

We have, fa(x) = Lak Ak xk-1 e - 2 2 > 0 rere) If O. /A, the Fisher information is given as, E la log tox (x) 2 IO) Ido we get Thus, separametrizing the pdf in terms of o, tolz- x2 -210 Gk rik (o oso R-1 е x>0 since d = yo Then log to(x) = -logla R logo (k-1) loga e logtala) + 2! 02 & + 20 es El blog tole) ER CS t2 02

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حد ما 262 20 E. che k² X Yamma (krd) e។ Now, since EX R > E. X ko EX? k²er - Eo X² (kesek)o? 2 Then, I (O) Eo Claro 21 x 2 2biko 04 k² + (R² + R) 0² Оч + per oz لاح ICO) Scanned with CamScanner 0 CS 4