Stat homework

Question No. 1
Solution : (a) w = = - exponential distribution
Also x > , that implies x/β > 1 , > ln (1) ,w>0    
That implies , x = β
dx/dw=β
pdf of w is given by , f(w)= f(x)* dx/dw=f(β* β
= *β
= * β
=
= ; w>0
Mean =
(b): Let , ,………. , be random sample from f(x;
Then find the joint density function from x vector.
f(x; = = /
= * *
Then by fisher Nayman’s factorization theorem
= (T(x), * h(x)
Where (T(x), = * * ; h(x)=1
Therefore is sufficient statistic for .
(c): Maximum Likelihood Estimator of     
Likelihood function is given by
L = / = /()^
Taking log on both sides
log⁡L = n log + n* log( - ()
differentiate w.r.t and equate equation =0
= n * +n* log(-)
· = n = n -
Therefore mle of = =

(f): Method of moments
f(x;β,) = ; x>β
E(X)=
= dx
=
= β/(
Let E(X) = X
Therefore X = β /(
By solving we get
=
(ii). N=6 ,β =5, X = 8.55
By putting values...