. Suppose we choose a random sample of size n from a from a uniform (0, ? ), with pdf f X (x)= 1 ? for x?(0,?) . Note that E( X )=? 2 and Var ( X )= ? 2 12 .

a) (3) Consider an estimator T=min Xi , where min Xi is the smallest observation. Using f T (t)= n ? (1? t ? ) n?1 for x?(0 ,?) , determine E(T ) . Hint: use the substitution u=1? t ? with du=? 1 ? .

b) (1) Since T=min Xi is not unbiased for ? , how would you adjust T to construct an unbiased estimator?

\f