Problems
These problems serve a variety of purposes. One is practice with the ideas from class. Some problems provide counterexamples and illustrate the regularity conditions of theorems. Some problems introduce, without much explanation, ideas we won't have time to consider in class. The first 3 are review intended to let me see how well you explain things you understand and to make sure the basics are already there.
SET A
, 
and W=V-U. Express the event
U=j and W=k in terms of X and Y.
and 
and prove that the event
U=j and the event W=k are independent.
SET B
are iid real random variables with
density f . Let 
be the X 's arranged
in increasing order, that is, X_(1) is the minimum of 
and 
while 
is the maximum.
.
. Prove that 
is independent of 
.
.
.
are iid exponential. Let 
for i=1, 2 and 3.
.
.
.
. Prove from the
definition of density that the density of X is 
.
). After observing X a coin landing
Heads with probability p is tossed X times. Let Y be the number of
Heads and Z be the number of Tails. Find the joint and marginal distributions
of Y and Z.
be the bivariate normal density with mean 0,
unit variances and correlation 
and let 
be the standard
bivariate normal density. Let 
.
Show that p has normal margins but is not bivariate normal.
and 
. Let Z=X+Y.
Find the distribution of Z given X and that of X given Z.
SET C
.
and 
are independent.
is uniformly
distributed on 
.
is a Cauchy random variable.
SET E
are iid 
and
are iid 
.
and 
.
. Find UMVUE's of
and of 
. (You have already found
the UMVUE for 
.)
and 
are unknown but that you
know that 
. Prove there is no UMVUE for 
.
(Hint: Find the UMVUE if you knew 
with a known.
Use the fact that the solution depends on a to finish the proof.)
iid Poisson( 
). Find the
UMVUE for 
and for 
.
iid with
for 
. For n=1 and 2
find the UMVUE of 
.
(Hint: The expected value of any function of X is a power series in
divided by 
. Set this equal to
and deduce that two power series
are equal. Since this implies their coefficients are the same you can see what
the estimate must be. )
SET F
are independent Poisson(
)
variables. Find the UMP level 
test of 
versus 
and evaluate the constants for the case n=3 and
.
) distribution with shape
parameter 
known. Find the UMPU test of 
and
evaluate the constants for the case 
and 
.
SET G
are independent
random variables. (This is the
usual set-up for the one-way layout.)
and 
.
be the error sum of squares in the ith cell in the
previous question.
.
of the form 
in the sense of mean squared error.
are observed what is the MLE of 
?
for the usual one-way layout model,
that is, the model of the last two questions.
are iid with density
are the minimal sufficient statistics show
that setting 
and solving gives the
likelihood equations. (Note the connection to the method of moments.)
SET H
at 1. You may use the splus function
integ.romb (or any other function) found by attaching the directory
/home/math/lockhart/research/software/quadrature/.Data.
be
the characteristic function of X. Show that
are independent random variables such that
. Prove that
where 
is the standard normal density. You should use the previous
problem and Taylor expansion of the characteristic function around 0.
Also do the same thing using Sterling's formula.
SET I
are iid exponential(
).
, 
, and 
for n=10, 20 and 40.
; get numerical values for n=10, 20 and 40.
are iid with density f. Assume the median
of f is 0. In this problem you will study the asymptotic distribution of
the sample median. To simplify things you may assume that n=2m-1 is odd
so that the median is the mth order statistic.
be the number of X's
less than or equal to x. Express the event that 
in terms of the number 
.
?
, of 
can be
written as 
.
is asymptotically normal
with mean 0 and variance 
.
is asymptotically
normal with mean 0 and variance 
.
for all i and let 
. What happens to
the MLE of 
?
SET J
are independent random variables
and that 
are the corresponding values of some covariate.
Suppose that the density of 
is
where 
, and 
are unknown parameters.
and an approximate 95% confidence interval for
.
and a model 
where the 
's are iid 
and the 
's are iid 
.
a number of animals 
are treated with the corresponding dose of some drug. The
number dying at dose d is Binomial with parameter 
. A common model
for 
is 
and 
.
;
express LD50 as a function of 
and 
.
against dose.
are a sample of size n from the density
In the following question
the digamma function 
is defined by
and the trigamma
function 
is the derivative of the digamma function. From
the identity 
you can deduce
recurrence relations for the digamma and trigamma functions.
known find the mle for 
.
and 
are unknown what equation must be
solved to find 
, the mle of 
?
/home/math/lockhart/teaching/801/gamma.Use this data in the following questions. First take
and find
the mle of 
subject to this restriction.
and 
to
get method of moments estimates 
and 
for
the parameters.
.
SET K
for b > 0.
prior distribution on p
with 
and 
both positive find the posterior
distribution of p given X.
for n=1,100 and asymptotically. Provide
graphs.