Neatly hand write your solutions - marks will be assigned for presentation
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, X = {0, 1, 2, ...}.
What distribution can we use to model X? Write down its probability mass function (p.m.f.).
Suppose that we observe the number of diseased trees on n randomly chosen one-acre parcels, X1, X2, ..., XN. The random variables X1, X2, ..., XN can be assumed to be inde-pendent. Write down the JOINT probability mass function for X1, X2, ..., XN. Simplify this expression which is a function of λ and the X′s.
We are going to use the Method of Maximum Likelihood to estimate λ. Write down the Likelihood function L(λ).
Write down the Log-likelihood function ℓ(λ).
Write down the Score Function S(λ).
Derive the maximum likelihood estimate of λ.
Write down the Information Function I(λ).
Use the second derivative test to show that you have found a maximum.
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.
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?
According to genetic theory, there are three blood types MM, NM and NN which should occur in a very large population with probabilities, θ2, 2θ(1−θ) and (1−θ)2, where θ is the (unknown) gene frequency, 0 ≤ θ ≤ 1.
Suppose that in a random sample of size n = 10 from the population, there are f1, f2, and f3 of types MM, NM and NN respectively. What distribution can we assume for the frequencies, f1, f2, and f3? Write down its probability mass function.
Blood Type
MM
NM
NN
Total
Observed frequency
f1
f2
f3
n = 100
Probability
p1 = θ2
p2 = 2θ(1 − θ)
p3 = (1 − θ)2
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(b) What are the assumptions required for the distribution you assume in part (a)?
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Write down the Log-likelihood function as a function of θ, ℓ(θ). Write down the Score function, S(θ).
Find an expression for the maximum likelihood estimate of θ.
Suppose that f1 = 3, f2 = 4, and f3 = 3. Compute the maximum likelihood estimate of θ.
(f)
Compute the estimated probabilities for p1, p2 and p3 under this model. i.e.
ˆ
=
pI(θ), i
1, 2, 3.
(g)
ˆ
Compute the estimated expected frequencies under this model npI(θ) and compare them
with the observed frequencies.
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