Assignment 2. Due: Nov 10, 2020, 11:30pm
1. Let X1, . . . , Xn be a random sample from a geometric distribution that has pmf f(x|θ) =
(1−θ)
x
θ, x = 0, 1, 2 . . ., 0 < θ < 1, zero elsewhere. Show that Pn
i=1 Xi
is a sufficient statistic
for θ.
2. Let X1, . . . , Xn be a random sample from a Beta(θ, 5). Show that the product X1×· · ·×Xn
is a sufficient statistic for θ.
3. Write the pdf, 0 < x < ∞, 0 < θ < ∞,
zero elsewhere, in the exponential form. If X1, X2, ..., Xn is a random sample from this
distribution, find a complete sufficient statistic Y1 for θ and the unique function φ(Y1) of this
statistic that is the MVUE of θ. Is φ(Y1) itself a complete sufficient statistic?
5. Let X1, . . . , Xn be a random sample from a uniform distribution [0, θ], for some unknown
parameter θ > 0. Is X(n)
, the largest order statistic among all samples, a sufficient statistic
for θ?
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