I prepare my exam and I came across this question:

Let $ X_1, X_2, …, X_n $ be iid with PDF $ f (x) = frac {2x} { theta ^ 2} $ for $ 0 leq x leq theta. $ Find the MLE of $ theta $

So here's what I did:

$ L ( theta) = prod frac {2x_i} { theta ^ 2} I (0 leq x leq theta) $

$ = (2n bar {x}) ( frac {1} { theta ^ {2n}}) I (0 leq min (x_i) leq max (x_i) leq theta) $

$ = (2n bar {x}) ( frac {1} { theta ^ {2n}}) I (0 leq X _ {(1)} leq X _ {(n)} leq theta $

In order to find MLE, I have to maximize $ L ( theta) $, by choosing a small $ theta $. however, $ theta $ can not be smaller than $ X_n $. I therefore conclude that:

$ hat { theta} = max (x_i) $

Is my calculation and my conclusion correct?