STA5326 - Mathematical Statistics - II Homework # 3 1. If X 1 , X 2 ,..., X n is a random sample of size n from N ( , 2 ), , show that n n n 2 2 X j , X j or X , X j = = j 1 j 1 =j 1 is a sufficient statistic for . 2. If X1 ,X 2 ,...,X n is a random sample of size n with p.d . f . = f ( x) e ( x ) , x , show that X (1) is a sufficient statistic for . 3. Let X 1 , X 2 ,..., X n be i.i.d . r.v.' s from P( ) (i.e= . f ( x) e x x! = , x 0,1,...). Find the UMVU for e . 4. Let X 1 , X 2 ,..., X n be a random sample of size n from N ( , 2 ). Find the UMVU for 2 2 . 5. Let X 1 , X 2 ,..., X n be i.i.d . r.v.' s from the Gamma distribution with known and = (0,) unknown. Then show that the UMVU estimator of is U ( X 1 , X 2 ,..., X n ) = n 1 n X j =1 j and its variance attains the Cramer-Rao= bound. (X f ( x) 1 x 1e x , x > 0.) n ( ) 6. Let X 1 , X 2 ,..., X m and Y1 , Y2 ,..., Yn be two independent random samples with the same mean and known variances 12 and 22 respectively. Then show that for every c [0,1], U = c X + (1 c)Y is an unbiased estimator of . Also find the value of c for which the variance of U is minimum. 7. If X 1 , X 2 ,..., X n be i.i.d . r.v.' s from the negative exponential distribution with parameter (0, ), show that 1 X is the MLE of . 8. Let X 1 , X 2 ,..., X n be i.i.d . r.v.' s from P( ) (i.e= . f ( x) e x x! = , x 0,1,...). Find the MLE for e . 9. Let X Binomial (n, p ), i.e., f ( x | p ) = ( p d )2 . ( nx ) p x (1 p)n x and L( p, d ) = Let = ( p ) 1, for 0 < p < 1, be a priori p.d . f . of p. Given a sample of size one, find the Bayes estimate of p