Homework 2
Solutions
1. Suppose X1 , X2 , ., X25 are i.i.d. N (, 2 ) where and are appropriately dened
and assumed known.Construct a statistic that is a function of all of the available
random variables( X1 , ., Xn ) that is distributed
2
a) N (, )
25
b)
Hw1 Stat 3445 s2013
due at the beginning of class Thursday, Jan. 30
A. Suppose that Z has a standard normal distribution and Y = a + bZ, where b > 0. Find
the density f (y) of Y and identify the distribution of Y by name.
B. Suppose that X, Y are random v
Homework 2
Due: 2/4/2015
1. Suppose X1 , X2 , ., X25 are i.i.d. N (, 2 ) where and are appropriately dened
and assumed known.Construct a statistic that is a function of all of the available
random variables( X1 , ., Xn ) that is distributed
2
a) N (, )
25
HW 1
Solutions
1. LetX be distributed Gamma(, ) and let a > 0 be a constant. Find the distribution of U = aX.
Solution:
h(X) = aX
h1 (U ) = U/a
fU (u) = fX (h1 (u)
fX (h1 (u)
d 1
h (u) =
du
=
=
d 1
h (u)
du
(u/a) 1
1
(u/a)1 e  
()
a
u
1
1
u1 ( ) e
Hw1

Stat 3445 s2015
due at the beginning of class Thursday Jan. 29
Z has a standard normal distribution and Y: a*bZ,where
the density /(y) of Y and identify the distribution of Y by name.
B. Suppose that X, Y are independent random variables with densit
HwZ
 Stat 3445 s2015
due at the beginning of class Thursday, Feb. 5
Notes:
o Reminder: work must be shown to receive credit.
o Results derived in cla.ss or in the text may be used without including a proof.
o
'oldentify" means to give the family of the d
Homework 3
Solutions
1. Let X1 , X2 , ., X121 be i.i.d. (2, ) where > 0. Construct a statistic that is
approximately distributed N (0, 1)
Solution: We know that by the Central Limit Theorem
X 2
X 2
X E[X1 ]
= 11
=
V [X1 ]/n
2 2 /121
2 2
Is approximately n
Homework 4
Solutions
Book problems:
1. Problem 8.1 from the book (p 394)
Solution:
2. Problem 8.2 from the book (p 394)
Solution:
3. Problem 8.3 from the book (p 394)
Solution:
4. Problem 8.4 from the book (p 394)
Solution:
5. Problem 8.5 from the book (p
Homework 5
Due: 3/4/2015
Note: For problems where the 1 condence level is specied (i.e. 95% condence
level), you are to nd the appropriate percentiles. You can look them up in the correct
table in appendix 3 of the book. You may also be able to nd these t
Homework 4
Due: 2/18/2015
Book problems:
1. Problem 8.1 from the book (p 394)
2. Problem 8.2 from the book (p 394)
3. Problem 8.3 from the book (p 394)
4. Problem 8.4 from the book (p 394)
5. Problem 8.5 from the book (p 394)
6. Problem 8.6 from the book
1 NeymanPearson Lemma
1.1 Power of a test
Denition:
Let X1 , ., Xn be an observed set of data from a distribution that depends on one unkown
parameter , and suppose we have some sort of hypothesis test with a test statistic U
and corresponding rejection
1 Large Sample Testing
As Statisticians, the most common testing situation we will face are situations in
which the Central Limit theorem applies
The typical structure will be along the following lines:
H0 : = 0
Ha : = 1
U = X
RR = cfw_X > k
We can thi
1 Hypothesis Testing
1.1 Hypotheses
Comparing two possible states of reality.
One is typically the state that is widely accepted. This is often called the Null
Hypothesis
The other is the opposing, or alternative state that contradicts the null hypothes
1 Method of Moments Estimation
1.1 The Technique
Coming up with estimators can be challenging.
Multiple techniques have been deveoped for nding good estimators
One of these techniques is called the Method of Moments
By denition we have the kth moment
1 Suciency
1.1 Denition
so far, we have evaluated estimators using the concepts of Bias and Eciency
Another concept we may use to evaluate estimators is Suciency
Denition:
Let X1 , ., Xn be a random sample from a probability distribution with unknown pa
Homework 6
Due: 3/11/2015
Let X1 , ., Xn be i.i.d. N (a + b, 2 + c)
1. Assume that 2 is known. Construct a two sided CI for based on X1 , ., Xn
when:
a) a = 1, b > 0, c = 0
b) a > 0, b = 0, c = 0
c) a = 1, b = 0, c > 0
2. Assume that 2 is unkown. Contruct
Homework 7
Due: 3/25/2015
Book problems:
1. Problem 9.1 from the book (p 447)
2. Problem 9.2 from the book (p 447)
3. Problem 9.3 from the book (p 447)
4. Problem 9.6 from the book (p 447)
5. Problem 9.8 from the book (p 448)
6. From problem 9.1 from the
Homework 10
Due: 4/23/2014
1. Problem 10.2 from the book (p 494)
Solution:
2. Problem 10.4 from the book (p 494)
Solution:
3. Problem 10.5 from the book (p 495)
Solution:
1
4. Problem 10.6 from the book (p 495
Solution:
5. Problem 10.7 from the book (p 49
Homework 8
Solutions!
1. Let X1 , ., Xn be i.i.d. (, ). Show that U =
is known.
Solution:
n
i=1 Xi
is sucient for when
X1 , ., Xn i.i.d. (, )
n
L(x1 , ., xn , )
x
1
i
x1 e
i
()
i=1
=
=
e
n
x
i=1 i
(
n
1
x1
)n
() i=1 i
n
i=1 Xi
So, by the factorizati
Homework 11
Due: 4/29/2015
1. Problem 10.23 from the book (p 505)
2. Problem 10.25 from the book (p 505)
3. Problem 10.44 from the book (p 510)
4. Problem 10.46 from the book (p 512)
5. Problem 10.53 from the book (p 517)
6. Problem 10.57 from the book (p
Homework 7
SOLUTIONS!
Book problems:
1. Problem 9.1 from the book (p 447)
Solution:
2. Problem 9.2 from the book (p 447)
Solution:
3. Problem 9.3 from the book (p 447)
Solution:
1
4. Problem 9.6 from the book (p 447)
Solution:
5. Problem 9.8 from the book
Homework 9
SOLUTIONS!
Book Problems
1. Problem 9.69 from the book (p 475)
Solution:
2. Problem 9.71 from the book (p 475)
Solution:
3. Problem 9.74 from the book (p 475)
Solution:
4. Problem 9.79 from the book (p 476)
Solution:
5. Problem 9.82 parts a & b
Homework 9
Due: 4/8/2015
Book Problems
1. Problem 9.69 from the book (p 475)
2. Problem 9.71 from the book (p 475)
3. Problem 9.74 from the book (p 475)
4. Problem 9.79 from the book (p 476)
5. Problem 9.82 parts a & b from the book (p 481)
6. Problem 9.8
Homework 10
Due: 4/22/2015
1. Problem 10.2 from the book (p 494)
2. Problem 10.4 from the book (p 494)
3. Problem 10.5 from the book (p 495)
4. Let X1 , ., X100 be i.i.d. Bin(n, p) where n is known. Construct an appropriate
large sample test statistic and
Homework 8
Due: 4/1/2015
1. Let X1 , ., Xn be i.i.d. (, ). Show that U =
is known.
n
i=1 Xi
is sucient for when
2. Let X1 , ., Xn be i.i.d. N Binomial(r, p). Show that U =
p when r is known.
n
i=1 Xi
is sucient for
3. Let X1 , ., Xn be i.i.d. U (0, ). Sh
Homework 5
Solutions!
Note: For problems where the 1 condence level is specied (i.e. 95% condence
level), you are to nd the appropriate percentiles. You can look them up in the correct
table in appendix 3 of the book. You may laso be able to nd these tabl
Homework 6
SOLUTIONS!
Let X1 , ., Xn be i.i.d. N (a + b, 2 + c)
1. Assume that 2 is known. Construct a two sided CI for based on X1 , ., Xn
when:
a) a = 1, b > 0, c = 0
Solution:
(Let Z/2 be the 1 /2 percentile of the N (0, 1) distribution).
2
X N ( + b,
Homework 3
Due: 2/11/2015
1. Let X1 , X2 , ., X121 be i.i.d. (2, ) where > 0. Construct a statistic that is
approximately distributed N (0, 1)
2. Let X1 , X2 , ., X144 be i.i.d. Beta(, ) where > 0 and > 0. Construct a
statistic that is approximately distr
1 Relative Eciency
Now that we have covered some basics of Estimation, we are going to return to
evaluating estimators eectively
The rst step in evaluating estimators is to measure them in terms of qualities
that we want
We already covered this when we
1 Sample Size
In some cases, experiemtners would like the results of an experiment to be accurate
within a certain amount, say b > 0.
In the case of large sample experiments, this is an achievable request
With large samples we know that X will be our e
Sampling distributions and the Central Limit Theorem
Lecture 3
Panpan Zhang
University of Connecticut
panpan.zhang@uconn.edu
January 26, 2017
Panpan Zhang (UConn)
Chapter 7
January 26, 2017
1 / 16
Statistics
Objective
The objective of statistics is to mak
Functions of Random Variables
Lecture 1
Panpan Zhang
University of Connecticut
panpan.zhang@uconn.edu
January 16, 2017
Panpan Zhang (UConn)
Chapter 6
January 16, 2017
1 / 17
Summary
In Chapter 6, we learned how to deal with the transformation of
random va
Functions of Random Variables
Lecture 2
Panpan Zhang
University of Connecticut
panpan.zhang@uconn.edu
January 24, 2017
Panpan Zhang (UConn)
Chapter 6
January 24, 2017
1/7
Motivation
Many functions of random variables of interest in practice depend on
the