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= If n = 20 and x = 5, what is the estimate? p = (b) Is the estimator of part (a) unbiased? O Yes O No (c) If n = 20 and x = 5, 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.)
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= If n = 20 and x = 5, what is the estimate? p = (b) Is the estimator of part (a) unbiased? O Yes O No (c) If n = 20 and x = 5, 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.)
Chapter8: Sequences, Series,and Probability
Section8.7: Probability
Problem 4ECP: Show that the probability of drawing a club at random from a standard deck of 52 playing cards is...
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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 =
If n = 20 and x = 5, what is the estimate?
p=
(b) Is the estimator of part (a) unbiased?
Yes
No
(c) If n = 20 and x = 5, 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.)
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