I'm preparing a lesson with a student, and since he's not a major in maths, I'm looking for a better solution than mine, because it looks too "mathematical"
We have a sample (iid) where Xi follows a uniform distribution over , where is a parameter.
Previously, it was asked to find the mean and the variance of Xi. And the cdf of |Xi| (y/theta for y in [0,theta])
Question 3) asks for the cdf of M, which is , for
It's ok from here.
Then, question 4) asks for the mean of M.
What I did is taking the derivative of G, and then
I'm already concerned that this is a too complicated method... So if you have a better one, denote it (1)
Second part of question 4) asks k such that W=kM is an unbiased estimator for
Nothing magic here,
Third part of question 4) asks to show that W converges in mean square.
It is not mentioned that it converges to . So how can I explain to the boy that it should converge to theta ?
This is the most awful part, because what I did is to use the property that :
converges to a constant c and
But calculating the variance of W is a really ugly... So if you have a better solution, please denote it (2)
And if you think these are the only ways to solve the questions, just eat a T-bone steak for dinner