# Download PDF by Frangakis C. E.: Rejoinder to discussions on clustered encouragement designs

By Frangakis C. E.

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Additional resources for Rejoinder to discussions on clustered encouragement designs with individual noncompliance Bayesian i

Sample text

On the other hand, such an objection is rather formal since all calculations were of "discrete" character. To give an absolutely rigorous proof one may, first, carry out all calculations for k games and, after this, consider the case k —► oo. The reader may also not do this but simply wait for rigorous definitions in Part II. 2. Properties of expectation The first three properties immediately follow from the definition. 1. I f P { X ( w ) > r ( u ; ) } = l, t h e n E X > E y . 2. For any constant c we have EcX = c EX.

Let 4 % of students in a group have obtained excellent grades for all mathematical courses and 10 % have excellent grades for Probability Theory. What is the probability that a student chosen at random has only excellent grades in Mathematics if it is known that this student does have the excellent grade for Probability Theory? Let A = { the student has only excellent grades in Mathematics} and B = { the student has the excellent grade for Probability Theory}. 4. □ Example 2. Consider a family having two children one of which is known to be a boy.

It is easily checked that (1) holds and, therefore, the formal definition (1) is consistent with our feeling on the correspondence between independence and equal likelihood of all outcomes in the symmetric case. □ Example 3. Consider Example 4 from Appendix to Section 1 and introduce the events A = { the first selected ball is red} and B = {the second selected ball is red}. Clearly, P (A) = rxj' r. In view of symmetry, P (B) = P (A). The reader is invited to check this fact by calculating P (B) directly.