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
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 concen
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/(
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 M
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 wa
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 ) = 24x
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
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 t
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)
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 w
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/(
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 M
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
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
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) Sho
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 M
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 =
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 t
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 a
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 Poiss
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):