Читать книгу Informatics and Machine Learning. From Martingales to Metaheuristics онлайн

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14 2.14 A rare genetic disease has probability P(disease) = 0.0000001. Suppose you have a test for this condition with sensitivity that is perfect (SN = 1.0) and with specificity that is 99.99% correct (SP = 0.9999) (i.e. false positives occur with probability 0.0001). Is this test useful in practice?

15 2.15 Prove P(X,Y|Z) = P(X|Z) P(Y|X,Z).

16 2.16 Prove the Bonferroni inequality: P(X,Y) ≥ P(X) + P(Y) − 1.

17 2.17 Suppose you are on a game show (with Monty Hall), and you are given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what is behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to change your pick door to No. 2?” Is it to your advantage to switch your choice? Prove by tabulation of possibilities, and then prove using Bayes' Rule with appropriate choice of variables.

18 2.18 Prove E(Y/X) ≥ E(Y)/E(X).