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D: Omer has the disease being tested for. ⚫ Tj: Omer tests positive on the jth test for j = 1, . Let p = P(D) be the prior probability that he has the d 1. Assume for this part the test results are conditiona status. Let a = P(T; | D) and bo = P(T; | Dc] Find the posterior probability that Omer has the di all n of the n tests. Hint: Since ao does not depend on j (the index of lot since P(T; | D) P(T; | D) = a² for any test i 2. Suppose some people have a gene that makes ther medical test. Let G be the event that Omer has the that D and G are independent - that is, the gene susceptible to the disease.

D: Omer has the disease being tested for. ⚫ Tj: Omer tests positive on the jth test for j = 1, . Let p = P(D) be the prior probability that he has the d 1. Assume for this part the test results are conditiona status. Let a = P(T; | D) and bo = P(T; | Dc] Find the posterior probability that Omer has the di all n of the n tests. Hint: Since ao does not depend on j (the index of lot since P(T; | D) P(T; | D) = a² for any test i 2. Suppose some people have a gene that makes ther medical test. Let G be the event that Omer has the that D and G are independent - that is, the gene susceptible to the disease.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter9: Multivariable Calculus
Section9.3: Maxima And Minima
Problem 35E
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By hand solve and focus on parts 2 and 3. Show all steps of the equations. 

• D: Omer has the disease being tested for. ⚫ Tj: Omer tests positive on the jth test for j = 1, 2, ..., n. Let p = P(D) be the prior probability that he has the disease. 1. Assume for this part the test results are conditionally independent given Omer's disease status. Let a = P(T; | D) and b₁ = P(T; | Dc), where ao and bo don't depend on j. Find the posterior probability that Omer has the disease, given that he tests positive on all n of the n tests. Hint: Since ao does not depend on j (the index of a test result), the algebra simplifies a lot since P(T; | D) P(T; | D) = a² for any test indices i and j. The same holds for bo. 2. Suppose some people have a gene that makes them always test positive on this type of medical test. Let G be the event that Omer has the gene. Assume that P(G) = ½ and that D and G are independent — that is, the gene does not make you more or less susceptible to the disease. If Omer has the gene, he'll test positive on all n tests. If Omer does not have the gene, then the test results are conditionally independent given his disease status. Let a₁ = P(T; | D, GC) and b₁ = P(T; | Dc, GC), where a1 and bi don't depend on j. Now, suppose that Omer tests positive on all n tests and find the posterior probability that Omer has the disease. 3. Using the same setup as in part (2.), find the posterior probability that Omer has the gene given that he tests positive on all n of the tests.
Transcribed Image Text:• D: Omer has the disease being tested for. ⚫ Tj: Omer tests positive on the jth test for j = 1, 2, ..., n. Let p = P(D) be the prior probability that he has the disease. 1. Assume for this part the test results are conditionally independent given Omer's disease status. Let a = P(T; | D) and b₁ = P(T; | Dc), where ao and bo don't depend on j. Find the posterior probability that Omer has the disease, given that he tests positive on all n of the n tests. Hint: Since ao does not depend on j (the index of a test result), the algebra simplifies a lot since P(T; | D) P(T; | D) = a² for any test indices i and j. The same holds for bo. 2. Suppose some people have a gene that makes them always test positive on this type of medical test. Let G be the event that Omer has the gene. Assume that P(G) = ½ and that D and G are independent — that is, the gene does not make you more or less susceptible to the disease. If Omer has the gene, he'll test positive on all n tests. If Omer does not have the gene, then the test results are conditionally independent given his disease status. Let a₁ = P(T; | D, GC) and b₁ = P(T; | Dc, GC), where a1 and bi don't depend on j. Now, suppose that Omer tests positive on all n tests and find the posterior probability that Omer has the disease. 3. Using the same setup as in part (2.), find the posterior probability that Omer has the gene given that he tests positive on all n of the tests.
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