Description
1. Does the Multinomial distribution, having pmf
f(x1, . . . , xk) =
n
x1 . . . xk
Y
k
j=1
p
xj
j
belong to an exponential family? Is the number of parameters k or k − 1?
2. Verify whether or not the following distributions belong to an exponential family. If they do, specify
the natural sufficient statistic and the natural parameter.
a. The Beta(α, β) distribution.
b. The Rayleigh distribution, having pdf
f(x) = 2
θ
x exp
−x
2
/θ
, x > 0, θ > 0.
c. The Weibull distribution, having pdf
f(x) = β
α
x
β−1
exp n
−x
β
/αo
, x > 0, α > 0, β > 0.
Consider separately the case when β is considered known and the case when it is not.
B. The following problems will be graded.
3. Consider a sample of size n from the Unif (0, θ) distribution. Assume a Pareto prior distribution for
θ, having pdf
p(θ) = αβα
θ
α+1 , θ ≥ β,
where α and β are positive constants (sometimes called hyper -parameters).
a. Show that the posterior distribution of θ has again a Pareto distribution, specifying the formulas
for updating the parameters α and β based on the observations.
b. Verify that the posterior distribution of θ based on the full data set is the same as the posterior
based on the distribution of the sufficient statistic T = max{Xi
: i = 1, . . . , n}.
4. Let X1, . . . , Xn
iid∼ N (µ, 1).
a. Find a one-dimensional sufficient statistic T and find its distribution.
b. Let I(µ) be Fisher information for the original model, and let IT (µ) be Fisher information for
the reduced model determined by T. Show that I(µ) = IT (µ).
