MAST20005/MAST90058: Week 3 Problems
Let X1,…,Xn be a random sample from N(μ,σ2) where −∞ < μ < ∞ and σ2 > 0. Assume that σ2 is known (i.e. it is a fixed, known value). Show the maximum likelihood estimator of μ is μˆ = X ̄.
(b) A random sample X1 , . . . , Xn of size n is taken from a Poisson distribution with mean λ > 0.
i. Show the maximum likelihood estimator of λ is λˆ = X ̄.
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ii. Suppose with n = 40 we observe 5 zeros, 7 ones, 12 twos, 9 threes, 5 fours, 1
five, and 1 six. What is the maximum likelihood estimate of λ?
(c) Let X1, . . . , Xn be random samples from the following probability density functions.
ˆ In each case find the maximum likelihood estimator θ.
i. f(x|θ)= 1 xexp(−x/θ), 0
7. Let X1,…,Xn be iid observations from X ∼ N(μ,σ2). Since X has a symmetric pdf, we might expect that both the sample mean X ̄ and the sample median πˆ0.5 will be good estimators of the population mean μ.
(a) Find the variance of X ̄.
(b) In general, the sample median will approximately follow a normal distribution, πˆ0.5 ∼ N (π0.5, π/2 × σ2/n), where π0.5 is the true median (we will learn more about this later in the semester). How does the variance of the sample median compare with that of the sample mean?
(n.b. π/2 is just the usual mathematical constant π divided by 2, it is not the same as the median π0.5.)
(c) Are the estimators biased?
(d) Which estimator do you expect to be more accurate?
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