Transformations
Feng-Chang Lin
Department of Biostatistics
University of North Carolina at Chapel Hill
flin@bios.unc.edu
(C&B 2.1, 4.3)
Lin (UNC-CH)
Bios 661
January 8, 2015
1 / 21
Introduction
In this unit we will learn how to answer the following sorts
BIOSTATISTICS 661
Bahjat Qaqish
Unit 5
Data Reduction
Reading assignment: C&B, Chapter 6
Introduction
Suppose that we are interested in a parameter . Further, suppose that we obtain a random sample,
say X, where the pdf or pmf of X is either completely kn
BIOSTATISTICS 661
Bahjat Qaqish
Unit 6
Point Estimation
Reading assignment: C&B, Chapter 7, sections 7.1, 7.2.1-2, 7.3.1-3.
Introduction
Random sample X1 , , Xn from f (x|)
: scalar or vector
Want: an estimator of , or of ().
Example: n(, 2 ), = (, 2 ) es
Bios 661/673
Homework 2
Bios 661: 1 5;
section
SP2015
Bios 673: 2 6(a)(b). Problem 6(c) will be solved in the discussion
1. C&B 5.22
There are more than one approaches. The most easiest one is as follows.
FZ (t) = 1 P (Z > t) = 1 P (X > t, Y > t) = 1 P (X
BIOSTATISTICS 661
Bahjat Qaqish
Transformations
Reading assignment: C&B, Sections 2.1, 4.3.
Introduction
In this unit we will learn how to answer the following sorts of questions:
1. X is Poisson(). Find the distribution of X 2 .
2. X exponential(). Find
Bios 661/673/SP2015
Bios 661: 1 5;
Homework 1
Feng-Chang Lin
Bios 673: 3 7
1. C&B 5.3
One can see Yi follows a Bernoulli distribution with mean p = P (Xi > ) = 1FX (u)
for every i. Since Y1 , . . . , Yn are i.i.d, the distribution of n Yi follows binomial
Data Reduction
Feng-Chang Lin
Department of Biostatistics
University of North Carolina at Chapel Hill
flin@bios.unc.edu
(C&B 6)
Lin (UNC-CH)
Bios 661
January 27, 2015
1 / 24
Introduction
Suppose that we are interested in a parameter .
Suppose that we obta
BIOSTATISTICS 661
Bahjat Qaqish
Unit 2
Random Samples
Reading assignment: C&B 5.1-5.5
Introduction Statistical inference is concerned with two entities; a population and a sample. The
overall picture is that a sample drawn from a population is used to mak
BIOSTATISTICS 661
Bahjat Qaqish
Unit 3
Ordrer Statistics
Reading assignment: C&B, Section 5.4
Introduction
One particular transformation of a random sample is to order the sample values in ascending order.
The ordered values are called the order statistic
BIOSTATISTICS 661
Bahjat Qaqish
Unit 4
Convergence Concepts
Reading assignment: C&B, Section 5.5
Introduction We studied earlier how to compute the distribution of data summaries and statistics
such as X and S 2 . For random samples from the normal distri
Bios 661/673/SP2015
Bios 661: 1 5;
Homework 1
Feng-Chang Lin
Bios 673: 3 7
6. Let X1 , . . . , Xn constitute a random sample of size n(n 3) from the parent population
fX (x) = ex , 0 < x < , 0 < <
(a) Find the conditional density function of X1 , . . . ,
Order Statistics
Feng-Chang Lin
Department of Biostatistics
University of North Carolina at Chapel Hill
flin@bios.unc.edu
(C&B 5.4)
Lin (UNC-CH)
Bios 661
January 20, 2015
1 / 12
Introduction
One particular transformation of a random sample is to order the
Bios 661/673
Bios 661: 1 5;
section.
Homework 9
SP2015
Bios 673: 2 6. Problem 7 will be discussed in the Bios 673 discussion
1. C&B 8.37
2. C&B 9.3
3. Let X1 , . . . , X8 be a random sample of size 8 from a Poisson distribution with mean
. To test the hyp
Bios 661/673
Bios 661: 1 5;
section.
Homework 6
SP2015
Bios 673: 2 6. Problem 7 will be discussed in the Bios 673 discussion
1. C&B 7.40
2. C&B 7.46
3. Let X1 , . . . , Xn be iid random variables from the N(, ) distribution, > 0.
(a) Calculate the Cramr-R
Random Samples
Feng-Chang Lin
Department of Biostatistics
University of North Carolina at Chapel Hill
flin@bios.unc.edu
(C&B 5.1-5.3)
Lin (UNC-CH)
Bios 661
January 13, 2015
1 / 23
Introduction
Statistical inferences are concerned with two entities; a popu
Interval Estimation
Feng-Chang Lin
Department of Biostatistics
University of North Carolina at Chapel Hill
flin@bios.unc.edu
(C&B 9)
Lin (UNC-CH)
Bios 661
April 2, 2015
1 / 23
Introduction
Example 1 Suppose X1 , . . . , Xn are iid from N(, 1).
We know tha
Point Estimation
Feng-Chang Lin
Department of Biostatistics
University of North Carolina at Chapel Hill
flin@bios.unc.edu
(C&B 7)
Lin (UNC-CH)
Bios 661
February 10, 2015
1 / 34
Introduction
Random sample X1 , , Xn from f (x|), where is either a scalar
or
Bios 661/673
Midterm 1 Practice
SP2015
If you are looking for practice questions, the following problems in the textbook are in
the right level for our midterm 1. You may nd the solution manual online and check the
solutions if you encounter diculties.
1.
Bios 661/673
Midterm 2 Practice
SP2015
If you are looking for practice questions, the following problems in the textbook are in
the right level for our midterm. You may nd the solution manual online and check the
solutions if you encounter diculties.
1. C
Bios 661/673/SP2015
Bios 661: 1 5;
Homework 1
Feng-Chang Lin
Bios 673: 3 7
1. C&B 5.3
2. C&B 5.8
3. C&B 5.13
4. Let X be a random variable distributed as standard normal (normal with mean 0
and variance 1). Find the pdf of |X| by two methods: using cdfs a
Bios 661/673
Bios 661: 1 5;
section
Homework 2
SP2015
Bios 673: 2 6(a)(b). Problem 6(c) will be solved in the discussion
1. C&B 5.22
2. C&B 5.23
3. C&B 5.25
4. (X, Y ) follows a bivariate normal distribution such that both X and Y have standard
normal mar
Bios 661/673
Bios 661: 1 5;
section.
Homework 3
SP2015
Bios 673: 2 6. Problem 7 will be solved in the Bios 673 discussion
1. C&B 5.32
2. C&B 5.35
3. Suppose that iid random variables X1 , . . . , Xn follow a uniform distribution on the
interval (0, 1) wit
Bios 661/673
Bios 661: 1 5;
section.
Homework 8
SP2015
Bios 673: 2 6. Problem 7 will be discussed in the Bios 673 discussion
1. C&B 8.20
2. C&B 8.31
3. Let X1 , . . . , Xn be random sample of size n having a pdf of the form f (x|) = 1/,
0 < x < , and zero
Bios 661/673
Bios 661: 1 5;
section.
Homework 5
SP2015
Bios 673: 2 6. Problem 7 will be discussed in the Bios 673 discussion
1. C&B 7.10
2. C&B 7.12
3. Suppose X1 , . . . , Xn are iid with pdf f (x|) = 2x/2 , 0 < x , zero elsewhere.
(a) Find MLE for .
(b)
Convergence Concepts
Feng-Chang Lin
Department of Biostatistics
University of North Carolina at Chapel Hill
flin@bios.unc.edu
(C&B 5.5)
Lin (UNC-CH)
Bios 661
January 22, 2015
1 / 25
Introduction
For random samples from the normal distribution, we derived