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Unformatted text preview: 36-226 Summer 2010 Homework 3 Due July 8 1. Let X1 , . . . , Xn be iid U(0, 2θ). Find the maximum likelihood estimator (MLE) for θ, θM LE .
Then ﬁnd E[θM LE ].
2. Let X1 , X2 , . . . , Xn be a random sample from the following distribution: fX (x) = αλα
xα+1 [λ,∞) with α, λ > 0. Show that the MLE for the pair (α, λ) is: n
(α, λ) = n
, X(1) .
ln( X i )
(1) Comment: This distribution is called the Pareto distribution, and it has been used in
economics as a model for a density function with slowly decaying tail.
3. The number of accidents per week in a certain intersection has a Poisson distribution with an
unknown parameter λ. In order to estimate λ, the intersection was observed for 10 randomly
chosen weeks. The only data that is available, though, is that in 2 out of the 10 weeks there
were no accidents in the intersection.
Based on this information only ﬁnd the maximum likelihood estimate for λ. 1 ...
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- Summer '09