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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?

Respuesta :

Answer:

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:

              [tex]f ( x; p , r ) = \left[\begin{array}{c}x+r-1&r-1\\\end{array}\right]*p^r*(1-p)^x[/tex]

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

             [tex]Ln |f ( x; p , r ) | = Ln |\left[\begin{array}{c}x+r-1&r-1\\\end{array}\right]| +r*Ln|p| + x*Ln|(1-p)|[/tex]  

- Derivative with respect to p on both sides:

             [tex]d (Ln |f ( x; p , r ) |)/dp = 0 +r/p - x/(1-p)\\\\Where , Ln |\left[\begin{array}{c}x+r-1&r-1\\\end{array}\right]| = Constant\\\\\\d (Ln |f ( x; p , r ) |)/dp = \frac{r - (r+x)*p}{p*(1-p)}[/tex]

- Set the derivative to zero and we get:

             [tex]0 = \frac{r - (r+x)*p}{p*(1-p)}\\\\r - (r+x)*p = 0\\\\p = \frac{r}{r+x}[/tex]

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

                     

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