Chapter 6, Problem 32SE

1. a. Let X₁,.., X_n be a random sample from a uniform distribution on [0, θ]. Then the mle of θ is $\hat{\theta }$ = Y = max(X_i). Use the fact that Y ≤ y iff each X_i ≤ y to derive the cdf of Y. Then show that the pdf of Y = max(X_i) is

   $f_Y(y)\text{ = }\{\begin{array}{l}\frac{n{y}^{n-1}}{{\theta }^n}\text{ 0 }\le y\le \theta \\ 0\text{ otherwise}\end{array}$

2. b. Use the result of part (a) to show that the mle is biased but that (n + 1 )max(X_t)//j is unbiased.