Answer:

And then the maximum occurs when
, and that is only satisfied if and only if:

Explanation:
For this case we have a random sample
where
where
is fixed. And we want to show that the maximum likehood estimator for
.
The first step is obtain the probability distribution function for the random variable X. For this case each
have the following density function:

The likehood function is given by:

Assuming independence between the random sample, and replacing the density function we have this:

Taking the natural log on btoh sides we got:

Now if we take the derivate respect
we will see this:

And then the maximum occurs when
, and that is only satisfied if and only if:
