TAILIEUCHUNG - Class Notes in Statistics and Econometrics Part 2

CHAPTER 3 Random Variables. . Notation Throughout these class notes, lower case bold letters will be used for vectors and upper case bold letters for matrices, and letters that are not bold for scalars. The (i, j) element of the matrix A is aij , and the ith element of a vector b is bi | CHAPTER 3 Random Variables . Notation Throughout these class notes lower case bold letters will be used for vectors and upper case bold letters for matrices and letters that are not bold for scalars. The i j element of the matrix A is aj and the ith element of a vector b is bg the arithmetic mean of all elements is b. All vectors are column vectors if a row vector is needed it will be written in the form bT. Furthermore the on-line version of these notes uses green symbols for random variables and the corresponding black symbols for the values taken by these variables. If a black-and-white printout of the on-line version is made then the symbols used for random variables and those used for specific values taken by these random variables can only be distinguished 63 64 3. RANDOM VARIABLES by their grey scale or cannot be distinguished at all therefore a special monochrome version is available which should be used for the black-and-white printouts. It uses an upright math font called Euler for the random variables and the same letter in the usual slanted italic font for the values of these random variables. Example If y is a random vector then y denotes a particular value for instance an observation of the whole vector yi denotes the ith element of y a random scalar and yi is a particular value taken by that element a nonrandom scalar . With real-valued random variables the powerful tools of calculus become available to us. Therefore we will begin the chapter about random variables with a digression about infinitesimals . Digression about Infinitesimals In the following pages we will recapitulate some basic facts from calculus. But it will differ in two respects from the usual calculus classes. 1 everything will be given its probability-theoretic interpretation and 2 we will make explicit use of infinitesimals. This last point bears some explanation. You may say infinitesimals do not exist. Do you know the story with Achilles and the turtle They are racing the

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