CHARACTERISTIC ROOTS AND VECTORS
1. A DIGRESSION
ON
COMPLEX NUMBERS
1.1. Denition of a complex number. A complex number is an ordered pair of real numbers denoted by (x1, x2 ). The rst member, x1, is called the real part of the complex number; the second
RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS
1. DISCRETE
RANDOM VARIABLES
1.1. Denition of a Discrete Random Variable. A random variable X is said to be discrete if it can assume only a nite or countable innite number of distinct values. A discrete rand
LARGE SAMPLE THEORY
1. ASYMPTOTIC EXPECTATION
AND
VARIANCE
1.1. Asymptotic Expectation. Let cfw_X n = X1, . . . , Xn , . . . be a sequence of random variables and let cfw_E Xn = E(X1), . . . , E(Xn), . . . be the sequence of their expectations. Suppose
n
1 September 2004 Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model A. Introduction and assumptions The classical normal linear regression model can be written as (1) or (2) where xtN is the tth row of the matr
MULTIVARIATE PROBABILITY DISTRIBUTIONS
1. PRELIMINARIES 1.1. Example. Consider an experiment that consists of tossing a die and a coin at the same time. We can consider a number of random variables dened on this sample space. We will assign an indicator r
1 September 2004 Statistical Inference in the Classical Linear Regression Model A. Introduction In this section, we will summarize the properties of estimators in the classical linear regression model previously developed, make additional distributional a
7 October 2004 Large Sample Properties of Estimators in the Classical Linear Regression Model A. Statement of the classical linear regression model The classical linear regression model can be written in a variety of forms. Using summation notation we wri
Restricted Least Squares, Hypothesis Testing, and Prediction in the Classical Linear Regression Model A. Introduction and assumptions The classical linear regression model can be written as (1) or (2) where xtN is the tth row of the matrix X or simply as
TRANSFORMATIONS OF RANDOM VARIABLES
1. INTRODUCTION 1.1. Denition. We are often interested in the probability distributions or densities of functions of one or more random variables. Suppose we have a set of random variables, X1, X2 , X3 , . . . Xn , with
1 September 2004 The Classical Linear Regression Model A. A brief review of some basic concepts associated with vector random variables Let y denote an n x l vector of random variables, i.e., y = (y1 , y2, . . ., yn )'. 1. The expected value of y is defin