A vehicle with a particular defect in its emission control system is taken to a succession of randomly selected mechanics until r = 18 of them have correctly diagnosed the problem. Suppose that this requires diagnoses by 20 different mechanics (so there were 2 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?

the likely-hood estimator of p is = r / (r+x)

Step-by-step explanation:

Given:

- The succession of randomly selected mechanics until r = 18

- The total number of diagnostics by different mechanics n = 20

Find:

What is the mle of p?

Solution:

- Let X denote the number of incorrect diagnosis. The random variable X follows a negative binomial distribution with parameters r and p.

- The likely-hood function can be defined as:

- Apply natural logs on both sides of the likely hood function:

- Derivative with respect to p on both sides:

- Set the derivative to zero and we get:

- Thus, the likely-hood estimator of p is = r / (r+x)

.there can be infinite values of m and n that can satisfy the above equation. for each value of m of m or n. there maybe infinite sets of solution for the above

step-by-step explanation: