Math 304: Homework 7
1. Let X1,...,Xnbe a random sample from a geometric distribution that has pmf
f(x;θ) = (1 −θ)xθfor x= 0,1,..., 0< θ < 1
(a) Show that Y=Pn
i=1 Xiis a sufficient statistic for θ.
(b) Show that the sufficient statistic is a function of the MLE.
2. Let X1,...,Xnbe iid N(0, θ).
(a) Show that Y=Pn
i=1 X2
iis a sufficient statistic for θ.
(b) What is the distribution of Y?
(c) Find an unbiased estimator based on the sufficient statistic.
(d) Is the estimator you found in the previous part the MVUE? (Why?)
3. Show that the nth order statistic of a random sample of size nfrom a U(0, θ) is a sufficient
statistic for θ. Generalize this result by considering the pdf
f(x;θ) = Q(θ)M(x),0< x < θ, 0< θ < ∞.
4. Let X1,...,Xnbe a random sample from a distribution with pdf
f(x;θ) = θ2xe−θx 0< x < ∞, θ > 0.
(a) Show that fis a regular case of the exponential family.
(b) Find Y, a complete sufficient statistic for θ.
(c) Compute E[1/Y ] and find a function of Ywhich is the unique MVUE of θ.
5. Let X1,...,Xnbe a random sample of size nfrom a distribution with pdf f(x;θ) = θxθ−1,0<
x < 1θ > 0.
(a) Show that the geometric mean (X1.. . Xn)1/n is a complete sufficient statistic for θ.
(b) Find the MLE of θand observe that it is a function of the geometric mean.
6. Let X1,...,Xnbe a random sample from a distribution that is N(θ, 1),−∞ < θ < ∞. Find
the MVUE of θ2.Hint: First determine E[¯
X2].
7. Let X1,...,Xndenote a random sample from a Poisson distribution with parameter θ > 0.
Find the MVUE of P(X≤1) = (1 + θ)e−θ.Hint: Let u(x1) = 1 if x1≤1, zero elsewhere,
and find E[u(X1)|Y=y], where Y=Pn
iXi.
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