I have an exam on 28th May. It is an introduction to probability and statistics.Topics are Limit Theorems Classical Statisical Inference.number of questions are 4. 30 minutes each question

a)
Estimate of theta
We get E(x/theta) = theta
Hence theta is expected value of X/theta
i.e. Also X =
Hence we get
b) We get Mean squared error
= = E(theta-theta)^2 ~0
i.e. variance sum of squares is kept to minimum
Qno.2
a) We are given f(xi) =
Multiply all functions of x1 x2… xn and take log
When we multiply we get
Product = , xi>=0
Take log to get
Take derivative wrt to theta
We get , x_i>0,
When we find second derivative we would get negative thus confirming maximum
b) To check whether this MLE is consistent.
For this we must show that when n tends to infinity this tends to theta
We know...