Obviously the MLE of $\theta$ for a distribution $X_1, X_2, \dots, X_n \sim Uniform(0,\theta)$ is $\hat{\theta} = max(X_1, X_2,\dots,X_n)$
Now, assume $\theta = 1$. If you take repeated samples with $n=50$. What would the distribution of $\hat{\theta}$ be?
I assume it would be: $$f(x) = P(\hat{\theta} = x) = P(X_1 \le x) P(X_2 \le x) \dots P(X_{50} \le x) = x^{50}$$
given that $x≤1$ always since
However, if you integrate this distribution, it does not sum to 1: $$\int_0^1x^{50}dx = \frac{1}{51}$$
so, would the "true" pdf be $$f(x) = 51x^{50}$$ or would it be a different function altogether?