A vehicle with a particular defect in its emission control system is taken to a

Question

A vehicle with a particular defect in its emission control system is taken to a succession of randomly selected mechanics until r = 2 of them have correctly diagnosed the problem. Suppose that this requires diagnoses by 20 different mechanics (so there were 18 incorrect diagnoses). Let p = P(correct diagnosis), so p is the proportion of all mechanics who would correctly diagnose the problem. What is the mle of p? p = Is it the same as the mle if a random sample of 20 mechanics results in 2 correct diagnoses? Explain.

No, the formula for the first one is (number of failures)/(number of trials) and the formula for the second one is (number of successes)/(number of trials). No, the formula for the first one is (number of failures)/(number of trials) and the formula for the second one is (number of successes)/(number of failures).
Yes, both mles are equal to the fraction (number of successes)/(number of failures).
No, the formula for the first one is (number of successes)/(number of failures) and the formula for the second one is (number of failures)/(number of trials).
Yes, both mles are equal to the fraction (number of successes)/(number of trials).

How does the mle compare to the estimate resulting from the use of the unbiased estimator p = r 1 /r + x 1 ?
The mle is greater than the the unbiased estimator.
The mle is equal to the the unbiased estimator.
The mle is less than the the unbiased estimator.

Explanation / Answer

Solution

(a) Yes, both mles are equal to the fraction (number of successes)/(number of trials).

(b) The mle is equal to the the unbiased estimator.

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