Ok, I'm using the notation from Maximum likelihood - Wikipedia, the free encyclopedia. So
is the joint PDF of the RV the n components in your sample.
We want to find, for given values of the value of that we will call that maximizes .
That is the idea.
What is the joint PDF? Figure out what the PDF for 1 component's lifetime as a function of (I add the extra 0 so we don't get confused). So , and thus the pdf of Y is .
We need to write this is a function of . Well as you pointed out
So , and so the pdf of Y is
Now at this point you could go on to write the joint PDF which is easy since the components are independent. Then maximize it (or the log of it).
However, we just wrote down or more to the point . We know MLE's have the functional invariance property (see Maximum likelihood). Since we know that the MLE for is , then is the MLE for .
Done. We didn't actually have to do much (since we kind of bootstrapped). But it helps to start to go through the motions of doing it outright since 1) we see how to set up the problem and 2) in doing so, we had to write down the formula that made it easy.