are iid uniform observations on the interval 
,
. Find
a minimal sufficient statistic for 
.
be iid observations from a location-scale family. Let R be the sample range
and S be the sample standard deviation. Verify that R/S is an ancillary
statistic.
be iid 
,
both 
and 
unknown. Consider a set of transformations 
,
where 
.
Show that 
is
a group.
,
independent, 
.
Find a UMP size-.05 test that 
against 
.
be a sequence independent random variables uniformly distributed on the
interval (0, 1). Let 
be the maximum of the first n observations. Determine the asymptotic distributions
of 
and 
.
be the Bayes estimator (posterior mean) of 
.
Show that
is unbiased only if 
.
be iid random variables from Poisson(λ), λ>0. Find the UMVU estimator
of λ.
be iid observations from a geometric distribution with parameter .
Let 
be the MLE of .
confidence interval for .---END---