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A diagnostic test for a certain disease is applied to n individuals known to not have the disease. Let X - the number among then test results that are positive (indicating presence of the disease, so X is the number of false positives) and p- the probability that a disease-free individual's test result is positive (i.e., p is the true proportion of test results from disease-free individuals that are positive). Assume that only X is available rather than the actual sequence of test results. (a) Derive the maximum likelihood estimator of p. p- 0.12 If n= 25 and x- 3, what is the estimate? p- 0.12 (b) Is the estimator of part (a) unbiased? O Yes O No (c) If n- 25 and x- 3, what is the mle of the probability (1 - p)5 that none of the next five tests done on disease-free individuals are positive? (Round your answer to four decimal places.) 0.5277

A diagnostic test for a certain disease is applied to n individuals known to not have the disease. Let X - the number among then test results that are positive (indicating presence of the disease, so X is the number of false positives) and p- the probability that a disease-free individual's test result is positive (i.e., p is the true proportion of test results from disease-free individuals that are positive). Assume that only X is available rather than the actual sequence of test results. (a) Derive the maximum likelihood estimator of p. p- 0.12 If n= 25 and x- 3, what is the estimate? p- 0.12 (b) Is the estimator of part (a) unbiased? O Yes O No (c) If n- 25 and x- 3, what is the mle of the probability (1 - p)5 that none of the next five tests done on disease-free individuals are positive? (Round your answer to four decimal places.) 0.5277

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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A diagnostic test for a certain disease is applied to n individuals known to not have the disease. Let X = the number among the n test results that are positive (indicating presence of the disease, so X is the number of false positives) and p = the probability that a disease-free individual's test result is positive (i.e., p is the true proportion of test results from disease-free individuals that are positive). Assume that only X is available rather than the actual sequence of test results. (a) Derive the maximum likelihood estimator of p. p = 0.12 If n = 25 and x = 3, what is the estimate? 0.12 (b) Is the estimator of part (a) unbiased? O Yes O No (c) If n = 25 and x = 3, what is the mle of the probability (1 – p)5 that none of the next five tests done on disease-free individuals are positive? (Round your answer to four decimal places.) 0.5277
Transcribed Image Text:A diagnostic test for a certain disease is applied to n individuals known to not have the disease. Let X = the number among the n test results that are positive (indicating presence of the disease, so X is the number of false positives) and p = the probability that a disease-free individual's test result is positive (i.e., p is the true proportion of test results from disease-free individuals that are positive). Assume that only X is available rather than the actual sequence of test results. (a) Derive the maximum likelihood estimator of p. p = 0.12 If n = 25 and x = 3, what is the estimate? 0.12 (b) Is the estimator of part (a) unbiased? O Yes O No (c) If n = 25 and x = 3, what is the mle of the probability (1 – p)5 that none of the next five tests done on disease-free individuals are positive? (Round your answer to four decimal places.) 0.5277
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