Homework 5 – STAT 543
On-campus: Due Friday, February 23 by 5:00 pm (TA’s office);
you also may turn in the assignment in class on the same Friday;
Distance students: Due Friday, March 2 by 12:00 pm (TA’s email)
1. Problem 6.2, Casella and Berger (2nd Edition)
2. Problem 6.3, Casella and Berger (2nd Edition)
3. Problem 6.5, Casella and Berger (2nd Edition)
4. example of Rao-Blackwell theorem; this is mostly a STAT 542 problem with STAT 543 implications.
Let X1 and X2 be iid Bernoulli(p), 0 < p < 1.
(a) Show S = X1 + X2 is sufficient for p.
(b) Identify the conditional probability P (X1 = x|S = s); you should know which values of x, s
to consider.
(c) Find the conditional expectation T ≡ E(X1 |S), i.e., as a function of the possibilities of S.
Note that T is a statistic.
(d) Show X1 and T are both unbiased for p.
(e) Show Varp (T ) ≤ Varp (X1 ), for any p.
5. Problem 6.21(a)-(b), Casella and Berger (2nd Edition)
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