Let $X_1, X_2, ..., X_n$ be independent and identically distributed uniform $U(0, \theta)$ distribution where $0 < \theta < \infty $. Show that $T(X)=max X_i$ is a complete statistics.

My biggest problem here is how to find the uniform distribution? Is it either:

a) The pdf is $f(x; \theta )$=$ \frac{1}{ \theta -0 }$=$ \frac{1}{ \theta }$

or

b) $f(x; \theta )$=$ \frac{1}{ \theta_2 - \theta_1 }$=$ \frac{1}{ \sqrt{3 \theta} + \sqrt{3 \theta } }$ = $ \frac{1}{ 2 \sqrt{3 \theta} }$ because the mean is $0$ and variance $ \theta $, where I find the value of $ \theta_1 $ and $ \theta_2 $ from it.

Then to find the complete statistics, I need to find the likelihood function and because it's need to show the maximum value, so I need to find it using order statistics $f_{Y_n}$

But, I can't continue since I'm not sure with my pdf of Uniform distribution.

Really appreciated if anyone could clear up the way to find the pdf of the Uniform distribution?