1. ## Introductory Statistics Questions

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

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

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

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

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

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

2. The joint density is

$f(x_1,\cdots , x_n)={1\over \theta ^n} I(0

where I is the indicator functions that either 0 or 1.

NOW switch to the order stats and you can factor this as....

$f(x_1,\cdots , x_n)={1\over \theta ^n} I(0

So the only stat stuck with theta is $X_{(n)}$ our suff stat.