C HAPTER
13
The Analysis of Variance
13.1
Introduction
13.2
The Analysis of Variance Procedure
13.3
Comparison of More Than Two Means: Analysis of Variance
for a One-Way Layout
13.4
An Analysis of Variance Table for a One-Way Layout
13.5
A Statistical Mod
C HAPTER
11
Linear Models
and Estimation
by Least Squares
11.1
Introduction
11.2
Linear Statistical Models
11.3
The Method of Least Squares
11.4
Properties of the Least-Squares Estimators: Simple Linear Regression
11.5
Inferences Concerning the Parameters
C HAPTER
1
What Is Statistics?
1.1 Introduction
1.2 Characterizing a Set of Measurements: Graphical Methods
1.3 Characterizing a Set of Measurements: Numerical Methods
1.4 How Inferences Are Made
1.5 Theory and Reality
1.6 Summary
References and Further R
C HAPTER
8
Estimation
8.1
Introduction
8.2
The Bias and Mean Square Error of Point Estimators
8.3
Some Common Unbiased Point Estimators
8.4
Evaluating the Goodness of a Point Estimator
8.5
Condence Intervals
8.6
Large-Sample Condence Intervals
8.7
Selecti
ANSWERS
Chapter 1
2.45 2.65, 2.65 2.85
7/30
16/30
Approx. .68
Approx. .95
Approx. .815
Approx. 0
y = 9.79; s = 4.14
k = 1: (5.65, 13.93); k = 2: (1.51,
18.07); k = 3: (2.63, 22.21)
1.15 a y = 4.39; s = 1.87
b k = 1: (2.52, 6.26); k = 2: (0.65,
8.13); k =
C HAPTER
2
Probability
2.1
Introduction
2.2
Probability and Inference
2.3
A Review of Set Notation
2.4
A Probabilistic Model for an Experiment: The Discrete Case
2.5
Calculating the Probability of an Event: The Sample-Point Method
2.6
Tools for Counting S
C HAPTER
9
Properties of Point
Estimators and Methods
of Estimation
9.1 Introduction
9.2 Relative Efciency
9.3 Consistency
9.4 Sufciency
9.5 The RaoBlackwell Theorem and Minimum-Variance Unbiased Estimation
9.6 The Method of Moments
9.7 The Method of Maxi
C HAPTER
6
Functions of
Random Variables
6.1 Introduction
6.2 Finding the Probability Distribution of a Function of Random Variables
6.3 The Method of Distribution Functions
6.4 The Method of Transformations
6.5 The Method of Moment-Generating Functions
6
C HAPTER
10
Hypothesis Testing
10.1
Introduction
10.2
Elements of a Statistical Test
10.3
Common Large-Sample Tests
10.4
Calculating Type II Error Probabilities and Finding the Sample Size for Z Tests
10.5
Relationships Between Hypothesis-Testing Procedur
C HAPTER
3
Discrete Random
Variables and Their
Probability Distributions
3.1
Basic Denition
3.2
The Probability Distribution for a Discrete Random Variable
3.3
The Expected Value of a Random Variable or a Function of a Random Variable
3.4
The Binomial Pro
C HAPTER
4
Continuous Variables
and Their Probability
Distributions
4.1
Introduction
4.2
The Probability Distribution for a Continuous Random Variable
4.3
Expected Values for Continuous Random Variables
4.4
The Uniform Probability Distribution
4.5
The Nor
PROBLEM SET 2
1) Suppose that X is a random variable that has the probability function
0.3 = 8
P X = k = p k = 0.2 = 10
0.5 = 6
a) What is the expected value of X?
b) What is the expected value of X2 ?
c) What
ECON 205 PROBLEM SET 7
1) The pdf of X is given as
6 1 0 < < 1
() =
0
a) Find the pdf of the random variable Y which is defined as = 2 + 1
b) Find the probability density of Y=X3
2) Let X and Y be two independent random variables with pdfs
! = ! > 0
Country
Austria
Austria
Austria
Austria
Austria
Austria
Austria
Austria
Austria
Austria
Austria
Belgium
Belgium
Belgium
Belgium
Belgium
Belgium
Belgium
Belgium
Belgium
Belgium
Belgium
Bulgaria
Bulgaria
Bulgaria
Bulgaria
Bulgaria
Bulgaria
Bulgaria
Bulgaria
ECON 205 PS 4
1) For two continuous variables and with probability densitiy functions f1 and f2
respectively, the random variable Y with probability density function
= ! + 1 ! ()
for some constant 0 1 is
10 The F distribution and its uses
10.1 Introduction
In section 5.2 we discussed the sampling distribution of the means
of samples taken from the same population, and in section 7.2 we
extended the concept of sampling distribution to the difference
betw
Gross Domestic
Gross Domestic Product
at Current Market Prices
% annual change at current prices
Gross Domestic Product at Constant Market Prices 2005, Chain Linking Method (Euro mn)
% annual change at constant prices
Percentage Annual Change of
Gross Dom
ECON 205 HW1 Solutions
1.
There is 36 possible outcome.
11 of the 36 outcomes gives Z=1
If X=1 and Y=1, then Z=1
If X=1, and Y=2,3,4,5,6 then Z=1
If Y=1, and X=2,3,4,5,6 then Z=1
9 of the 36 ou
ECON 205 HW 3 (Due to May 25th)
1)
Suppose X is distributed uniformly on the interval [0,1]. A random
variable defined as U=3X+1. Find the probability density function (pdf) and
probability distribution function of
ECON 205 HW 1
Due Date: March 27th 2015, 17:00 (CHAPTERS 2&3)
1) Two balanced dice are rolled, one is green and one is red. Let X be the
value facing up on the green die and Y be the value facing up on the red die.
Let Z = min cfw_X,Y, that is, th
PROBLEM SET 2
1) Suppose that X is a random variable that has the probability function
0.3 = 8
P X = k = p k = 0.2 = 10
0.5 = 6
a) What is the expected value of X?
b) What is the expected value of X2 ?
c) What i
ECON 205 PS 4
1) For two continuous variables and with probability densitiy functions f1 and f2
respectively, the random variable Y with probability density function
= ! + 1 ! ()
for some constant 0 1 is
ECON 205 Problem Set 3
1) Let Y be a discrete random variable with mean and variance 2. If
a and b are constants, prove that;
a) E(aY+b)=aE(Y)+b=a+b
b) Var(aY+b)=a2Var(Y)= a22
2) Let k 1 be a positive integer a
ECON 205 Problem Set 5
4. Let Y1 and Y2 have the joint probability density function given by
6 1 ! , 0 ! ! 1
! , ! =
0,
a) Show that this is a probability density function.
Find
b) The marginal densi
ECON 205 PS4 Solutions
1.
b)
2.
3.
4.
5.
6. Let Y be the daily water demand in city X in the early afternoon. Then, Y is exponential
with = 100 cfs.
7. We want to find the capacity a so that ( ) = 0.01. For 1, we have
1
( ) = 5(1 )4 = (1 )5
5
We need ( )
Introduction The Expected Value of a Random Variable Moments Chebyshevs Theorem Moment Generating Functions Product
Lecture 5: Mathematical Expectation
Assist. Prof. Dr. Emel YAVUZ DUMAN
MCB1007 Introduction to Probability and Statistics
Istanbul K
ult
ur
ECE 316 Probability Theory and Random Processes
Chapter 5 Solutions (Part 2)
Xinxin Fan
Problems
13. You arrive at a bus stop at 10 oclock, knowing that the bus will arrive at some time uniformly
distributed between 10 and 10 : 30.
(a) What is the probabi
Chapter 3:
Consumer Preferences
and the Concept of Utility
Outline
Introduction
Description of consumer preferences
The Utility functions
Marginal utility and diminishing marginal utility
Indifference curves
Marginal rate of substitution
Special function
Microeconomics
3070 Prof. Barham
Lecture 1: Introduction
Syllabus and Website
Website:
http:/www.colorado.edu/ibs/hb/barham/courses/eco
n3070/
All assignments and solution keys will be posted on
the web site.
I will send you a notice when they are poste
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