Hello,

Okay, I don't know at all if I did these questions correctly...

Let

and where

Let X be a rv with pdf f.

We know that

Let be a sample of iid rvs following the same distribution as X.

Let the rv and let

Preliminary questions : we proved that Y/n is an unbiased and convergent estimator for .

We also proved that Y follows a binomial distribution with parameter

First question : calculate an estimator T for by the method of moments.

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So for this one, ... We know that

So we can take , right ?

I thought of using observations of but then there is a table of values for , not . So I considered it was too easy...

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Second question : prove that T converges in probability to and calculate its bias

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So in order to prove that it converges, I used the LLN (the one stating the convergence in probability), then used Slutsky's theorem.

Then for its bias, I find 0... is it normal ?? This is where I am the most doubtful...

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My own questions :

- is the estimator by the method of moments unique ?

- is it always an unbiased estimator ?

it may sound stupid, but I don't know how to be sure...

Thanks in advance !