STAT 409
Fall 2011
Homework #11
(due Friday, December 2, by 4:00 p.m.)
From the textbook:
8.7-1 (
)
8.4-2 (
8th edition (
8.7-3 (
)
)
8.4-4 (
)
8.7-4 (
)
)
8.7-6 (
8.4-10 (
)
)
_
8.
In Neverland, annual income (in $) is distributed according to Gamma dist
STAT 409
Fall 2012
Homework #2
( due Friday, September 14, by 4:00 p.m. )
1.
Let X 1 , X 2 , , X n be a random sample from the distribution with probability
density function
(
)
f X ( x ) = f X ( x ; ) = 2 + x 1 (1 x ) ,
a)
0 < x < 1,
> 0.
~
Obtain the m
MATH/STAT 409
Homework # 3
due 09/20/2013
1. Let > 0 and let X1 , X2 , . . . , Xn be a random sample of size n from a distribution
with pdf
f (x; ) =
43
x , 0 < x < .
4
(a) Find the MLE .
(b) Is a consistent estimator? Justify your answer.
(c) Is an unbia
STAT 409
Fall 2012
Name
Version A
ANSWERS
.
Exam 1
Page
Earned
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partial credit might depend on it.
1
Put your final answers at the end of
your work, and mark them clearly.
2
3
No credit will be given
without supporting w
MATH/STAT 409
Homework # 1
due 09/06/2013
1. Suppose = cfw_1, 2, 3 and the pdf f (x; ) is
= 1 : f (1; 1) = 0.6, f (2; 1) = 0.1, f (3; 1) = 0.1, f (4, 1) = 0.2
= 2 : f (1; 2) = 0.2, f (2; 2) = 0.3, f (3; 2) = 0.3, f (4, 2) = 0.2
= 3 : f (1; 3) = 0.3, f
MATH/STAT 409
Homework # 11
Due 12/06/2013
1.
7.2-8
2.
Metaltech Industries manufactures carbide drill tips used in drilling oil wells.
The life of a carbide drill tip is measured by how many feet can be drilled before
the tip wears out. Metaltech claims
MATH/STAT 409
Homework # 2
due 09/13/2013
1. 6.1-10
2. 6.1-18 (Hint: Use MoM)
3. Let X1 , X2 , . . . , Xn be a random sample from a distribution with pdf
fX (x; ) = ( + 1)(1 x) , 0 < x < 1, > 1.
Obtain the method of moment estimator of , .
4. Let X1 , X2
STAT 409
Fall 2012
Homework #4
( due Friday, September 28, by 4:00 p.m. )
1.
6.2-8 (
)
2.
6.2-12 (
)
3.
6.2-18 (
)
4.
6.4-4
The data in Exercise 6.2-12 that give results of a leakage test are repeated here:
3.1
3.3
4.5
2.8
3.5
3.5
3.7
4.2
3.9
3.3
Use thes
STAT 409
Fall 2012
Homework #4
( due Friday, September 28, by 4:00 p.m. )
1.
6.2-8 (
a)
x = 46.42;
b)
s 2 = 41.682,
)
s 6.456,
t 0.05 ( 4 ) = 2.132,
46.42 2.132 6.456
5
46.42 6.156
or
( 40.264, 52.576 ).
2.
6.2-12 (
)
a)
x = 3.580;
b)
s2 =
c)
t 0.05 ( 9 )
STAT 409
Fall 2012
Name
Version B
ANSWERS
.
Exam 2
Page
Earned
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partial credit might depend on it.
1
Put your final answers at the end of
your work, and mark them clearly.
2
3
No credit will be given
without supporting w
STAT 409
Fall 2012
Homework #11
(due Friday, December 7, by 4:00 p.m.)
1.
Alex has a special coin he uses to make decisions in class. We do not know whether
the coin is fair or not, so we assign the following prior probability distribution on the
probabil
STAT 409
Fall 2012
Homework #9
(due Friday, November 16, by 4:00 p.m.)
1.
Let X 1 , X 2 , , X 5 be a random sample of size
n=5
from a Poisson distribution
with mean > 0. We wish to test H 0 : = 2 vs. H 1 : 2.
n=5
Recall that Y =
Xi
has Poisson ( 5 ) dist
STAT 409
Fall 2012
Homework #1
( due Friday, September 7, by 4:00 p.m. )
1.
6.1-10 (
)
+
(d)
~1
p=
(a)
E( X ) = 1 p .
(b)
~
p equals the number of successes, n, divided by the number of Bernoulli
/
trials,
X = 1~
p
so
X
=n
n
i =1 X i
;
n
i =1 X i ;
(c)
STAT 409
Fall 2012
Homework #3
( due Friday, September 21, by 4:00 p.m. )
1.
Let > 0 and let X 1 , X 2 , , X n be a random sample from the distribution with
the probability density function
f X (x) = f X ( x ; ) =
a)
x
2
e x ,
x > 0.
Find the sufficient s
MATH/STAT 409
Homework # 5
Due 10/18/2013
1.
6.4-4
The data in Exercise 6.2-12 that give results of a leakage test are repeated here:
3.1
3.3
4.5
2.8
3.5
3.5
3.7
4.2
3.9
3.3
Use these data to find a 95% confidence interval for
2.
(a)
(b)
.
(
MATH/STAT 409
Homework # 6
Due 10/18/2013
1.
6.4-4
The data in Exercise 6.2-12 that give results of a leakage test are repeated here:
3.1
3.3
4.5
2.8
3.5
3.5
3.7
4.2
3.9
3.3
Use these data to find a 95% confidence interval for
(a)
2.
2.
(b
STAT 409
Fall 2011
Homework #6
( due Friday, October 14, by 4:00 p.m. )
1.
In the past, only 30% of the people in a large city felt that its mass transit system is
adequate. After some changes to the mass transit system were made, we wish to test
if the p
STAT 409
Fall 2012
Name _
Extra Credit
(3 points)
No credit will be given without supporting work.
1.
Use mathematical induction to prove that
13 + 23 + 33 + n3 =
n 2 (n + 1) 2
4
for all positive integers n.
Base.
Step.
13 =
n = 1.
Suppose
3
1 2 (1 + 1 )
STAT 409
Fall 2011
Homework #5
( due Friday, October 7, by 4:00 p.m. )
1.
Consider Gamma ( , = usual ) distribution. That is,
f ( x; , ) =
1
( )
x 1 e x
,
0 < x < .
Suppose is known.
a)
Determine the Fisher information I ( ).
x
ln f ( x; ) = ln ( ) ln +
STAT 409
Fall 2012
Homework #10
(due Friday, November 30, by 4:00 p.m.)
1 2.
Let > 0 and let X 1 , X 2 , , X n be independent random variables, each with
the probability density function
f (x; ) =
n
Recall
Y=
ln X i =
i =1
1.
n
i =1
x 1.
,
W i has Gamma
STAT 409
Fall 2012
Homework #7
(due Friday, October 26, by 4:00 p.m.)
1.
A researcher wishes to determine whether the starting salaries of high-school math
teachers in private schools are higher than those of high-school math teachers in
public schools. S
STAT 409
Fall 2015
A. Stepanov
Version
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ANSWERS
A
.
NetID _
Exam 2
Page
Earned
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partial credit might depend on it.
1
2
No credit will be given
without supporting work.
3
4
5
The exam is closed book and closed notes.
STAT 409
1 4.
Practice Problems 2
Fall 2015
A. Stepanov
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.
y
It follows that
+1
1
f Y ( y ) = k
,
y k;
> 0.
STAT 409
Fall 2012
Name
ANSWERS
.
Version B
Quiz 4
(10 points)
Be sure to show all your work, your partial credit might depend on it.
No credit will be given without supporting work.
1.
Province Ranch is an insurance company that provides homeowners polic
STAT 409
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:
Fall 2015
A. Stepanov
Practice Problems 3
ln x
x
n
from the distribution with
x > 1,
,
> 1.
X 1
~ 2
If > 2, the method of m
STAT 409
Fall 2016
A. Stepanov
Examples for 10/31/2016
H0: = 0
H 1 : = 1.
vs.
Likelihood Ratio:
( x 1 , x 2 ,., x n ) =
L ( 0 ; x 1 , x 2 ,., x n )
.
L ( 1 ; x 1 , x 2 ,., x n )
Neyman-Pearson Theorem:
C = cfw_ ( x 1 , x 2 , , x n ) : ( x 1 , x 2 ,., x n
STAT 409
Fall 2016
A. Stepanov
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.
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Version A
Quiz 3
(8 points)
Be sure to show all your work; your partial credit might depend on it.
No credit will be given without supporting work.
1. (8) Let > 1 and let X 1 , X 2 , , X n be a random
STAT 409
Fall 2016
A. Stepanov
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ANSWERS
.
NetID _
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Quiz 1
(8 points)
Be sure to show all your work; your partial credit might depend on it.
No credit will be given without supporting work.
1. (8) Let > 1 and let X 1 , X 2 , , X n be a random
STAT 409
Fall 2016
A. Stepanov
Name
ANSWERS
.
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Quiz 2
(8 points)
Be sure to show all your work; your partial credit might depend on it.
No credit will be given without supporting work.
1.
Let > 1 and let X 1 , X 2 , , X n be a random samp
Exercises 4E5
Let
_ . 0 Fn<3Ci . (i S 07
[t1- (r) :2 g i
' Emit) v d. it) -4 r! > 0.
and K ,_ .
g limit + ('27. Fail) if i.
F r : .- - >
L'\ L l.- FHH 7 / Z (J
The two-step functions 5(1) and FAX) yield a 100(1 _ a)% confidence band for
the unknown distri
STAT 409
Fall 2015
A. Stepanov N&lll6 ANSWERS
Version A NSllD
Exam 1
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Be sure to Show all your work; your
1 partial credit might depend on it.
2 No credit will be given
3 without supporting work.
4 The exam is closed book
STAT 409
1.
Fall 2015
A. Stepanov
Practice Problems 5
Let > 0 and let X 1 , X 2 , , X n be independent random variables, each with
the probability density function
f (x; ) =
x2
,
a)
Obtain the maximum likelihood estimator of , .
b)
Is a consistent estimat
STAT 409
1 4.
Let > 0 and let X 1 , X 2 , , X n be a random sample from the distribution
with the probability density function
f X (x) = f X ( x ; ) =
2
e x ,
x
n
Recall:
i =1
Xi
has Gamma ( = n, usual =
1
x > 0.
) distribution.
We wish to test H 0 : = 5
STAT 409
From the textbook:
Practice Problems 10
Fall 2015
A. Stepanov
9th edition 8th edition
8.7-1
D = 0.40 < 0.45
Do NOT Reject H 0 at = 0.20.
at x = 0.40-
an 80% confidence band for F ( x ) is included ( note the CDF of U ( 0, 1 ) is inside )
8.7-3
D