Hi. I really need help with these statistics problem!!

Let $\displaystyle X_1, ..., X_n$ be a random sample from a uniform distribution over $\displaystyle (0, \theta)$.

a) Find the joint density $\displaystyle f(x_1, ... , x_n| \theta)$ and use the Factorization theorem to show $\displaystyle T = max(X_1, ..., X_n)$ is sufficient for $\displaystyle \theta$.

b) Find the distribution function of $\displaystyle T$ and hence show the p.d.f. of $\displaystyle T$ is $\displaystyle h(t)= n(\frac{t}{n})^{n-1} \frac{1}{\theta}$, $\displaystyle 0 < t < \theta$.

Thanks for your time

c) Find the conditional density $\displaystyle g(x_1,... , x_n|t, \theta$) and show this does not depend on $\displaystyle \theta$.

d) Show $\displaystyle E(T)= \frac{n}{n+1}\theta$ and hence find a function of the sufficient statistic that is an unbiased estimator of $\displaystyle \theta$.