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
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
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
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
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
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
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
C HAPTER
5
Multivariate Probability
Distributions
5.1
Introduction
5.2
Bivariate and Multivariate Probability Distributions
5.3
Marginal and Conditional Probability Distributions
5.4
Independent Random Variables
5.5
The Expected Value of a Function of Ran
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 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 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 ( )
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 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
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
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Belgium
Belgium
Belgium
Belgium
Belgium
Belgium
Belgium
Belgium
Belgium
Belgium
Belgium
Bulgaria
Bulgaria
Bulgaria
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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
PROBLEM SET 7
1) The pdf of X is given as
6(1 )
0< <1
() = cfw_
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
() = > 0
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
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
Middle East Technical University
Department of Economics
SPRING 2013-2014
ECON 205
Due Date: May 9th 2014, 17:00
HOMEWORK 2
1) X is a continuous random variable that follows the uniform distribution over the
interval [0,4].
(a) Draw the probability densit
Middle East Technical University
Department of Economics
SPRING 2013-2014
ECON 205
Due Date: April 7th 2014, 17:00
HOMEWORK I (CHAPTERS 2&3)
1) Two balanced dice are rolled, one is red and one is green. Let X be the value facing up
on the green die and Y
PROBLEM SET 7
1) The pdf of X is given as
f (x)= 6 x ( 1x ) 0< x <1
0 elsewhere
cfw_
a) Find the pdf of the random variable Y which is defined as
Y =2 X +1
b) Find the probability density of Y=X3
2) Let X and Y be two independent random variables with pdf