# Thread: Maximum Likelihood estimator question

1. ## Maximum Likelihood estimator question

Hi,

I'm stuck on this question...
I think I've got part a) and part b) I'm just struggling on how to justify it...
It's mainly parts c) and d) that I can't do as I don't really know how I'd go about finding the variance of an estimator, and then I don't fully understand what a sampling distribution is so any help would be much appreciated...

Thanks

2. First note that is exponentially distributed with mean
Then we immediately have and
(or you can prove it easily by integration)

a) For the M.L.E. part, I guess it is a standard work.
First write down the joint log-likelihood function:

Differentiate it with respect to the parameter and set it equal to 0:

b) To check the estimator is unbiased or not, you need to check the equality

which is obviously true in this case, because the sample mean is always an
unbiased estimator of the mean.

c) Again, by using the i.i.d. property of the random sample,

d) You should be able to prove that

either by moment generating function, or convolution and induction
Then you can do a simple transformation to obtain the p.d.f. of

Sorry if I give too detailed hints.

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