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Let X1,..., X, be a random sample from the following probability distribution x +1 fx (x) x > 0, 0 elsewhere, %3D e 0(0 + 1) and A

Let X1,..., X, be a random sample from the following probability distribution x +1 fx (x) x > 0, 0 elsewhere, %3D e 0(0 + 1) and A

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Can you please help solve the advanced statistics and probability theory question below?

parts (c) only please

how to Find its expected value and its variance of S?

I attached my answers for part a and b for reference

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Let X1,..., X, be a random sample from the following probability distribution x +1 0(0 + 1)` fx(x) x > 0, 0 elsewhere, %3D and 0> 0. (a) Show that f belongs to a regular exponential family. (b) Find a complete sufficient statistics S of 0. (c) Find its expected value and its variance of S.
Transcribed Image Text:Let X1,..., X, be a random sample from the following probability distribution x +1 0(0 + 1)` fx(x) x > 0, 0 elsewhere, %3D and 0> 0. (a) Show that f belongs to a regular exponential family. (b) Find a complete sufficient statistics S of 0. (c) Find its expected value and its variance of S.
Amsiver (@ A famidy of fodf is called Exponential Family if it can an be expressed as f(xle) = h(x) cco) exp(Q(O). T(X) Now -240 e f(a1@)= (X+1) • OCO+1) exF{-6 Comparing with (1) hex) = (x+1) we get C(O) = TeR) = X Hence f belongs family. to reqular exponential (6) ) To find the complete sufficient statistic S of o we use the following the orem . Theoremo fet X,, X2 , Xn be iid observations from an exponential family with odf of the form frx1o)= hcx) c@☺ exf ( Q0), TCX) Them S= E TCX) es complete sufficient statistic . As, fexl@) belongs to exponential family and TX) = 2 :- Complete suffricient statistic ΣΤΚ) T(X) 131 ズー complete suffecient statistic Therefore, A of0
Transcribed Image Text:Amsiver (@ A famidy of fodf is called Exponential Family if it can an be expressed as f(xle) = h(x) cco) exp(Q(O). T(X) Now -240 e f(a1@)= (X+1) • OCO+1) exF{-6 Comparing with (1) hex) = (x+1) we get C(O) = TeR) = X Hence f belongs family. to reqular exponential (6) ) To find the complete sufficient statistic S of o we use the following the orem . Theoremo fet X,, X2 , Xn be iid observations from an exponential family with odf of the form frx1o)= hcx) c@☺ exf ( Q0), TCX) Them S= E TCX) es complete sufficient statistic . As, fexl@) belongs to exponential family and TX) = 2 :- Complete suffricient statistic ΣΤΚ) T(X) 131 ズー complete suffecient statistic Therefore, A of0
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