*1.4 Realizable rules. This section, giving sucient conditions for randomized decision
rules to b e realizable, is relatively abstract and can b e passed over on rst reading.
A measurable space (Y, V ) will b e called a Lusin space if there exists a metri
March 21, 2003
1.5 The sequential probability ratio test. Sec. 1.1 treated tests b etween two laws
P and Q on a sample space X . If we have independent, identically distributed (i.i.d.)
observations X1 , . . . , Xn with distribution P or Q, then X can b e
18.466 midterm exam, Wednesday, April 2, 2003, 2:05-2:55 P. M.
Closed book exam. No books, notes, or calculators may be consulted during this
exam. (Calculators are irrelevant since no questions involve numerical calculation.)
1. (a) Dene the likelihood r
February 14, 2003
1.6 Sequential decision theory. As previously, let the sample space b e a measurable
space (X, B ) and P = cfw_P , a family of laws on it. Let X1 , X2 , . . . , b e independent
and identically distributed with law P . Let A b e the spac
Feb. 24, 2003
1.7 Proof of optimality of the SPRT. For the Neyman-Pearson Lemma, we had a
statement, Theorem 1.1.3, not involving losses or priors, and another (Theorem 1.1.8)
when there are losses and a prior. Here, the statement of Theorem 1.5.1 again d
18.466 review session May 14, 2003: excerpts of sections 3.3-4.1. Its important to
know all denitions given in this document. Several theorems are stated in simplied forms
without listing all their assumptions. In place of such assumptions its sucient to
NAME:
18.466 nal exam, Wednesday, May 21, 2003, 9 A.M.-noon
Closed book exam. No books or notes may be consulted during this exam.
There are 13 questions on the exam. Answer any TEN of the 13 for full credit. Please
indicate which three you omit.
Explanat