STAT 410
Fall 2011
Homework #5
(due Friday, October 7, by 3:00 p.m.)
1.
Every month, the government of Neverland spends X million dollars purchasing guns and
Y million dollars purchasing butter. Assume X and Y jointly follow a Bivariate Normal
distributio
STAT 410
Homework #12
(due Thursday, December 13, by 4:30 p.m.)
Fall 2012
1 2.
Let > 0, and let X 1 , X 2 , . , X n be a random sample from the distribution with the probability density function
f ( x; ) = 3 x 5 e - x
We wish to test
2
,
x > 0.
H0: =
1 1
STAT 409
Homework #10
( due Friday, November 30, by 3:00 p.m. )
Fall 2012
1.
Consider Gamma ( , = "usual " ) distribution. That is,
f ( x; , ) =
Suppose is known. a)
1 ( )
x -1 e - x
,
0 < x < .
Determine the Fisher information I ( ).
ln f ( x; ) = - ln
STAT 410 Fall 2012 Version C
Name
Earned
ANSWERS
.
Page 1
Exam 2
Be sure to show all your work; your partial credit might depend on it.
Put your final answers at the end of your work, and mark them clearly.
2
3
If the answer is a function, its support mus
STAT 410 U3, G4
Fall 2011
Homework #1
(due Friday, September2, by 3:00 p.m.)
1.
Below is a list of moment-generating functions. Provide (i) the values for mean
and variance 2 , and (ii) P ( 1 X 2 ) for the random variable associated with
each moment-gene
STAT 410
Fall 2011
Homework #9
(due Friday, November 4, by 3:00 p.m.)
1 - 2.
Let X 1 , X 2 , , X n be a random sample from the distribution with probability
density function
f X ( x ; ) = ( + 1 ) (1 x ) ,
1.
a)
n
i =1
> 1.
0 < x < 1,
Find a sufficient st
STAT 410
Fall 2011
Homework #7
(due Friday, October 21, by 3:00 p.m.)
0.
Warm-up:
4.2.3
By Chebyshevs Inequality,
P(|Wn | )
2
W
n
2
=
b
0
n p2
as n
for all > 0.
P
Wn .
Therefore,
1.
Suppose P ( X n = i ) =
n+i
, for i = 1, 2, 3.
3n + 6
Find the limiting
STAT 410
Fall 2011
Homework #6
(due Friday, October 14, by 3:00 p.m.)
1 - 4.
Let X 1 , X 2 , , X n be a random sample from the distribution with probability
density function
f X ( x ; ) = ( + 1 ) (1 x ) ,
1.
> 1.
0 < x < 1,
~
Obtain the method of moments
STAT 410
Fall 2011
Homework #8
(due Friday, October 28, by 3:00 p.m.)
1.
4.3.16
Hint:
a)
et
2
n = 1 + t + t + o 1 for large n.
n 2n n
M X ( t ) = ( 1 t ) 1,
1
(
t < 1.
)
t n Xn 1
t n t ( X 1 + X 2 + . + X n
=e
M Y ( t ) = E e
E e
n
=e
t
t
= e
b)
et
t
STAT 410 Fall 2012
Name
ANSWERS
.
Extra Credit 2
( 2 points )
No credit will be given without supporting work.
1.
Let X 1 , X 2 , . , X n be a random sample of size probability density function
n
from the distribution with
f X (x) = f X ( x ; ) = ( - 1 )
STAT 420
2GR, 2UG, 3GR, 3UG
X1
X2
X =
.
X
n
Examples for 01/21/2016
Spring 2016
A. Stepanov
1
2
E( X ) = X =
.
n
n-dimensional
random vector
Covariance matrix:
11 12
22
X = 21
.
.
n1 n 2
ij = Cov ( Xi , Xj )
. 1n
. 2n
.
. nn
X = E
STAT 420
2GR, 2UG, 3GR, 3UG
Spring 2016
A. Stepanov
Examples for 01/28/2016
The (normal) simple linear regression model:
Y i = 0 + 1 x i + i ,
where
is
are independent Normal ( 0 , 2 ) ( iid Normal ( 0 , 2 ) ).
0 , 1 , and 2 are unknown model parameters
An Introduction to R
Notes on R: A Programming Environment for Data Analysis and Graphics
Version 3.2.3 (2015-12-10)
W. N. Venables, D. M. Smith
and the R Core Team
This manual
Copyright c
Copyright c
Copyright c
Copyright c
Copyright c
is for R, version
STAT 420
2GR, 2UG, 3GR, 3UG
1.
a)
Fall 2014
A. Stepanov
Practice Problems 1
At Anytown College, the heights of female students are normally distributed with
mean 66 inches and standard deviation 1.5 inches. The heights of male students are
also normally d
STAT 410
1.
Summer 2016
A. Stepanov
Practice Problems 18
If the random variable Y denotes an individuals income, Paretos law claims that
P ( Y y ) = k , where k is the entire populations minimum income. It
y
follows that
+1
1
f Y ( y ) = k
,
y k;
> 0.
STAT 410
1.
Let X 1 , X 2 , , X n be a random sample of size
probability density function
f X (x) = f X ( x ; ) = ( 1 ) 2
Recall:
Summer 2016
A. Stepanov
Practice Problems 20
ln x
x
n
from the distribution with
x > 1,
,
> 1.
X 1
~ 2
If > 2, the method o
STAT 410
1.
Let X 1 , X 2 , , X n be a random sample from the distribution with the
probability density function
f X ( x ; ) = ( + 1 ) (1 x ) ,
a)
Summer 2016
A. Stepanov
Practice Problems 19
n
Y = ln ( 1 X i
i =1
Recall:
)
0 < x < 1,
> 1.
has Gamma ( =
STAT 410
Fall 2011
Homework #11
(due Friday, December 2, by 3:00 p.m.)
1.
Let X 1 , X 2 , X 3 , X 4 be a random sample of size
n=4
from a Geometric ( p )
distribution ( the number of independent trials until the first success ). That is,
P ( X 1 = k ) = (
STAT 410
Fall 2011
Homework #2
(due Friday, September 9, by 3:00 p.m.)
1.
Let X and Y have the joint p.d.f.
f X Y ( x, y ) = C x 2 y 3 ,
a)
0 < x < 1, 0 < y <
x,
zero elsewhere.
What must the value of C be so that f X Y ( x, y ) is a valid joint p.d.f.?
x
STAT 410
Fall 2011
Homework #12
(due Thursday, December 8, by 4:30 p.m.)
1 4.
Consider two continuous random variables X and Y with joint p.d.f.
40 x y 3
f X, Y ( x, y ) =
0
1.
a)
0 < x < 1, 0 < y < x 2
otherwise
Find P ( 4 X < 5 Y ).
P ( 5 Y > 4 X ) =
STAT 410 Fall 2012
Name
ANSWERS
.
Extra Credit
( 3 points )
No credit will be given without supporting work.
1.
In Alex's neighborhood, trick-or-tricking starts at 6:00 p.m., and the children would be showing up at Alex's door according to Poisson process
STAT 410 Fall 2012
Name
ANSWERS
.
Exam 1
( part 2 )
( 8 points )
No credit will be given without supporting work.
Put your final answers at the end of your work, and mark them clearly.
If the answer is a function, its support must be included. Do NOT use
STAT 410 Fall 2012 Version A
Name
ANSWERS
.
Exam 1
Page 1 2 3 4 5 6 The exam is closed book and closed notes. You are allowed to use a calculator and one 8" x 11" sheet with notes. Earned
Be sure to show all your work; your partial credit might depend on
STAT 410 Fall 2012 Version B
Name
ANSWERS
.
Exam 1
Page 1 2 3 4 5 6 The exam is closed book and closed notes. You are allowed to use a calculator and one 8" x 11" sheet with notes. Earned
Be sure to show all your work; your partial credit might depend on
STAT 410 Fall 2012
Name
ANSWERS
.
Exam 2
( part 2 )
( 8 points )
No credit will be given without supporting work.
Put your final answers at the end of your work in the space provided. If the answer is a function, its support must be included.
1.
Let > 0 a
STAT 410 Fall 2012 Version A
Name
Earned
ANSWERS
.
Page 1
Exam 2
Be sure to show all your work; your partial credit might depend on it.
Put your final answers at the end of your work, and mark them clearly.
2
3
If the answer is a function, its support mus
STAT 410 Fall 2012 Version B
Name
Earned
ANSWERS
.
Page 1
Exam 2
Be sure to show all your work; your partial credit might depend on it.
Put your final answers at the end of your work, and mark them clearly.
2
3
If the answer is a function, its support mus
STAT 410 Fall 2012 Version D
Name
Earned
ANSWERS
.
Page 1
Exam 2
Be sure to show all your work; your partial credit might depend on it.
Put your final answers at the end of your work, and mark them clearly.
2
3
If the answer is a function, its support mus
STAT 409
Homework #8
( due Monday, November 5, by 4:30 p.m. )
Fall 2012
1 2.
Let > 0 and let X 1 , X 2 , . , X n be independent random variables, each with the probability density function
f (x; ) =
^ Obtain the maximum likelihood estimator of , .
x2
,
x
STAT 410
Homework #9
(due Monday, November 12, by 4:30 p.m.)
Fall 2011
1.
Let X 1 , X 2 , . , X n be a random sample from the distribution with probability density function
f X ( x ; ) = ( + 1 ) (1 - x ) ,
Recall:
X
0 < x < 1,
> 1.
~ 1 = - 2 is the metho