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.

If n = 25 and x = 3, what is the estimate?

(b) Is the estimator of part (a) unbiased?

(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.)

Part a

The maximum likelihood estimator of p, is the sample proportion,  

And the proportion of the estimate at n=25 and x = 3  is 0.12.

Part b

The maximum likelihood estimate from part (a) is an unbiased estimator.

Part c

The maximum likelihood estimate of the probability is 0.5277.

Step-by-step explanation:

a)

The likelihood function is,

Applying log on both sides:

Find the differentiation of log L with respect to P and equate to zero to find the MLE for .

Calculate the estimate of the function if n=25 and x = 3.

Hint for next step

The MLE for the population proportion is the sample proportion and there is a 0.12 proportion of estimate at n = 25 and x = 3  

b)

Consider  

Since , the maximum likelihood estimate from part (a) is an unbiased estimator.

Hint for next step

Based on the part (a) calculation, since it is proved that , the maximum likelihood estimate is unbiased estimator for population proportion

(c)

Calculate the maximum likelihood estimate of the probability .

By the Invariance property of MLE, to find the MLE of , the MLE of p can be used.

That is, the MLE of is

Therefore,

MLE of the probability,

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