Final answer:
To find the maximum likelihood estimate of μ, we need to use the given information. We know that in a randomly selected chapter with 40 pages, there were no errors in the first 15 pages and at least one error in each of the remaining 25 pages. The likelihood of having no errors in the first 15 pages is (e^(-μ))^15 = e^(-15μ), and the likelihood of having at least one error in each of the remaining 25 pages is (1 - e^(-μ))^25. To find the maximum likelihood estimate, we need to differentiate the likelihood function, set the derivative equal to zero, and solve for μ.
Step-by-step explanation:
To find the maximum likelihood estimate of μ, we need to use the given information. We know that in a randomly selected chapter with 40 pages, there were no errors in the first 15 pages and at least one error in each of the remaining 25 pages.
Since the number of word-processing errors per page follows a Poisson distribution, the probability of no errors in a page with mean μ is given by P(X = 0) = e^(-μ). Therefore, the likelihood of having no errors in the first 15 pages is (e^(-μ))^15 = e^(-15μ).
The likelihood of having at least one error in each of the remaining 25 pages is 1 - P(X = 0) = 1 - e^(-μ). Therefore, the likelihood of having at least one error in each of the remaining 25 pages is (1 - e^(-μ))^25.
To obtain the maximum likelihood estimate, we need to find the value of μ that maximizes the likelihood function. Taking the product of the two likelihoods, we get: L(μ) = e^(-15μ) * (1 - e^(-μ))^25.
To find the maximum likelihood estimate of μ, we differentiate the likelihood function with respect to μ, set the derivative equal to zero, and solve for μ. We then plug in this value of μ into the likelihood function to get the maximum likelihood estimate.