MLE for the scrap rate on a line

Question

A quality check sorts each part into pass, minor defect, major defect, or scrap, with fixed probabilities summing to 1. Out of 1000 parts the counts are 700, 180, 90, 30.

By maximum likelihood, what is the estimated probability that a part is scrap?

Accepted Answer

The multinomial log-likelihood \sumj nj \ln pj maximized under \sumj pj = 1 gives, via a Lagrange multiplier, the observed frequencies: \hat pj = \frac{nj}{n}. For scrap, \hat p{	ext{scrap}} = \frac{30}{1000} = 0.03. Takeaway. Each category's MLE is its own count divided by the total, regardless of how many categories there are.

MLE for the scrap rate on a line

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