chapter 2.26 answer.doc - Homework Homework 1 1 M D...

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Homework Homework 1 1. M. D. Computing describes the use of Bayes’ theorem and the use of conditional probability in medical diagnosis. Prior probabilities of diseases are based on the physician’s assessment of such things as geographical location, seasonal influence, occurrence of epidemics, and so forth. Assume that a patient is believed to have one of two diseases, denoted 1 D and 2 D with 6 . 0 1 D P and 4 . 0 2 D P that medical research has determined the probability associated with each symptom that may accompany the diseases. Suppose that given diseases 1 D and 2 D , the probabilities that the patient will have symptoms 1 S , 2 S or 3 S are as follows. 1 S 2 S 3 S 1 D 0.15 0.1 0.15 2 D 0.80 0.15 0.03 After a certain symptom is found to be present, the medical diagnosis may be aided by finding the revised probabilities of each particular disease. Compute the posterior probabilities of each disease given the following medical findings. (a) The patient has symptom 1 S (b) The patient has symptoms 2 S or 3 S (c) For the patient with symptom 1 S in part (a), suppose we also find symptom 2 S . What are the revised probabilities of 1 D and 2 2. Let 2 1 , ~ , , , N X X X n , with 2 known. For the prior distribution for , we shall assume the normal distribution 2 , N (a) Find the maximum likelihood estimate. (b) Find the posterior distribution of and posterior mode as estimate of . (c) Compare the estimates in (a) and (b). and . . D . 2 . . 1
3. Let P X X X n ~ , , , 1 , ! | i x i x e x f i For the prior distribution for , we shall assume the gamma distribution e G 1 1 , , ~ . (a) Find the maximum likelihood estimate. (b)Find the posterior distribution of and posterior mean as estimate of . (c) Compare the estimates in (a) and (b). 2 . Solution 1: 3
676 . 0 222 . 0 15 . 0 ) | ( 3 2 3 2 1 3 2 1 S S P S S D P S S D P and 324 . 0 222 . 0 072 . 0 ) | ( 3 2 3 2 2 3 2 2 S S P S S D P S S D P . (c) We can use 1 1 | S D P and 1 2 | S D P as new revised probability for the diseases. That is, 2195 . 0 | reviesed , 1 1 1 S D P D P and 7805 . 0 | reviesed , 1 2 2 S D P D P .
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