density function of $X_{(n)}$ for a uniform $[0,\theta]$ distribution?

Question

I have to check whether the estimator $X_{(n)}$ is an unbiased estimator for the uniform $[0,\theta]$-distribution. But what is the density function of the value $X_{(n)}$?

Answer 1

$P(X_{(n)}\leq x)=P(X_1,\ldots X_n\leq x)=P(X_1\leq x)^n=(x/\theta)^n$, for $0\leq x\leq\theta$, where I use that the $X_i$'s are i.i.d. This gives the distributiom function for the maximal observation. Now differentiate.