STAT 830
Problems: Assignment 4
1. Suppose that Y1 , . . . , Yn are independent random variables and that
x1 , . . . , xn are the corresponding values of some covariate. Suppose
that the density of Yi is
f (yi ) = exp (yi exp( xi ) xi ) 1(yi > 0)
where ,
STAT 801
Assignment 1 Solutions
1. The concentration of cadmium in a lake is measured 17 times. The
measurements average 211 parts per billion with an SD of 15 parts per
billion. Could the real concentration of cadmium be below the standard
of 200 ppb?.
T
STAT 801
Solutions: Asst 2
1. Suppose X has the Beta(, ) density
f (x; , ) =
( + ) 1
x (1 x) 1 1(0 < x < 1)
()( )
Find the distribution of Y = X/(1 X ).
Solution: the inverse transformation is x = y/(1 + y ) for y = 1. The
derivative dx/dy is (1 + y )2 .
STAT 801
Problems: Assignment 5 1. Suppose cfw_Xij ; j = 1, . . . , ni ; i = 1, . . . , p are independent N (i , 2 ) random variables. (This is the usual set-up for the one-way layout.) (a) Find the MLE's for i and . (b) Find the expectations and variance
STAT 801
Problems: Assignment 1 This rst problem set is review. I want to see how you answer relatively elementary problems. I dont plan to discuss these with anyone before they are handed in and I want complete clear explanations about what you are doing
STAT 801
Problems: Assignment 2 1. Suppose X has the Beta(, ) density f (x; , ) = ( + ) -1 x (1 - x)-1 1(0 < x < 1) ()()
Find the distribution of Y = X/(1 - X). 2. In class I showed f (x1 , x2 ) = 24x1 x2 1(0 < x1 )1(0 < x2 )1(x1 + x2 < 1) is a density. I
STAT 801
Problems: Assignment 3 1. Suppose X1 , . . . , Xn are iid real random variables with density f . Let X(1) , . . . , X(n) be the X's arranged in increasing order. (a) Find the joint density of X(1) , . . . , X(n) . (b) Suppose f = 1[0,1] . Prove t
STAT 801
Solutions: Asst 4
1. Develop explicit formulas for the saddlepoint approximation to the density of the mean of a sample of size n from the exponential distribution.
Compare the results with the true Gamma density.
The cumulant generating function
STAT 801
Solutions: Asst 3
1. Suppose X1 , . . . , Xn are iid real random variables with density f . Let X(1) , . . . , X(n)
be the X s arranged in increasing order.
(a) Find the joint density of X(1) , . . . , X(n) .
Let g be the joint density and A = cf
STAT 801
Problems: Assignment 4 1. Develop explicit formulas for the saddlepoint approximation to the density of the mean of a sample of size n from the exponential distribution. Compare the results with the true Gamma density. 2. Suppose X, Y and Z are i
STAT 801
Solutions: Asst 2
1. Suppose X has the Beta(, ) density
f (x; , ) =
( + ) 1
x (1 x) 1 1(0 < x < 1)
()( )
Find the distribution of Y = X/(1 X ).
Solution: the inverse transformation is x = y/(1 + y ) for y = 1. The
derivative dx/dy is (1 + y )2 .
STAT 801
Problems: Assignment 5
1. Suppose cfw_Xij ; j = 1, . . . , ni ; i = 1, . . . , p are independent N (i , 2 ) random variables.
(This is the usual set-up for the one-way layout.)
(a) Find the MLEs for i and .
Solution:
i =
Xij /ni
j
and
(Xij i )2
i
STAT 801
Problems: Assignment 7 1. For the last problem on assignment 5 do a likelihood ratio test of Ho : = 1. 2. Suppose X1 , . . . , Xn are independent Poisson() variables. Find the UMP level test of 1 versus > 1 and evaluate the constants for the case
STAT 801
Problems: Assignment 6 1. Postponed from Assignment 5: Let Ti be the error sum of squares in the ith cell in the rst question of Assignment 5. (a) Find the joint density of the Ti . (b) Find the best estimate of 2 of the form error.
p 1
ai Ti in
STAT 830
Problems: Assignment 2
1. Let p1 be the bivariate normal density with mean 0, unit variances
and correlation and let p2 be the standard bivariate normal density.
Let p = (p1 + p2 )/2.
(a) Show that p has normal margins but is not bivariate normal
STAT 830
Problems: Assignment 5
1. Suppose cfw_Xij ; j = 1, . . . , ni ; i = 1, . . . , p are independent N (i , 2 )
random variables. (This is the usual set-up for the one-way layout.)
(a) Find the MLEs for i and .
(b) Find the expectations and variances
STAT 830
Problems: Assignment 3
NOTE: you only need to do problems 1 and 2 if you havent already done
them on assignment 2.
1. Suppose X and Y are independent with X N (, 2 ) and Y
N (, 2 ). Let Z = X + Y . Find the distribution of Z given X and that
of
STAT 830
Problems: Assignment 6
1. For the last problem on assignment 5 do a likelihood ratio test of Ho :
= 1.
2. From the text: page 171, # 5 a and b.
3. From the text: page 191, # 3.
4. From the text: page 191, # 5.
5. From the text: page 191, # 6a.
6
STAT 830
Problems: Assignment 1
This rst problem set is partly review. I want to see how you answer
relatively elementary problems. I dont plan to discuss these with anyone
before they are handed in and I want complete clear explanations about
what you ar
STAT 830
Problems: Assignment 2
1. Suppose X and Y have joint density f (x, y ). Prove from the denition
of density given in class that the density of X is g (x) = f (x, y ) dy .
2. Suppose X is Poisson(). After observing X a coin landing Heads with
proba
STAT 830
Problems: Assignment 4
1. Consider the empirical distribution funct Fn (x) for a sample X1 , . . . , Xn
from a cdf F . In this problem I want you to compare several condence
limits for F (x):
The pointwise interval based on the approximately nor