Question

Suppose that diseased trees are distributed randomly and uniformly throughout a large forest with an average of λ per acre. Let X denote the number of diseased trees in a randomly chosen one-acre plot with range, 0, 1,2,...^ (a) What distribution can we use to model X? Write down its probability mass function (p.m.f (b) Suppose that we observe the number of diseased trees on n randomly chosen one-acre parcels, Xi, X2, ..., Xn. The random variables X1, X2,..., Xn can be assumed to be inde- pendent. Write down the JOINT probability mass function for Xi, X2, ..., Xn. Simplify this expression which is a function of λ and the X's (c) We are going to use the Method of Maximum Likelihood to estimate λ. Write down the Likelihood function L(X) (d) Write down the Log-likelihood function e(A) (e) Write down the Score Function S(A) (f) Derive the maximum likelihood estimate of λ (g) Write down the Information Function I(X) (h) Use the second derivative test to show that you have found a maximum (i) Suppose that the numbers of diseased trees observed in eight randomly chosen one-acre parcels were: 5, 8, 9, 2, 10, 7, 6, 10, Compute the maximum likelihood estimate of λ using this data. (i) Suppose that the unit of measure was a five-acre plot, i.e. we found the same number of diseased trees in eight randomly chosen five-acre plots, but λ is still the mean number per one acre. What is the maximum likelihood estimate of λ now?

please help with all of these questions, thanks.

Answer 1

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