Nicholas A. Macri

Analytical and Empirical Study of the Tails of Probability Distributions

For the distribution function F(x) of a random variable X we define the end points

MATHEMATICS OF FINANCE

\alpha (F) = \inf \{ x:F(x) > 0\} {\text{and }}\omega {\text{(F) = sup\{ x:F(x) < 1\} }}

MATRICES AND LINEAR SYSTEMS

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COUNTING TECHNIQUES; PERMUTATIONS AND COMBINATIONS

Abstract In flow microfluorometry (FMF) analysis cells stained with a fluorescent dye that binds specifically to DNA are passed through the instrument. The number of cells in the population having a fluorescence intensity is recorded in a single channel of a multichannel pulse height analyzer. The result is a DNA fluorescence histogram for the population. A method for decomposing an FMF histogram into its G 1 , S and G 2 + M components, corresponding to similarly designated phases of the cell cycle is given. This technique can also be applied to find the parameters in all of the previous approaches. The parameters are calculated by iteration which eliminates the need for non-linear optimization procedures.

LINEAR PROGRAMMING (A GEOMETRIC APPROACH)

This chapter discusses the basic mathematics used in finance. If P = principal (amount borrowed), r = interest rate (a percentage per year), t = time (in years that the principal is held), and S = amount, then the simple interest formula is S = P (1 + rt ). If the number of annual conversions is imagined to increase indefinitely, then a situation is approached in which interest is compounded continuously. The continuous compound interest formula is given by S = Pe rt . The chapter also discusses the effective rate of interest formula, effective interest rate for continuous compounding, simple discount, the future value of an ordinary annuity, and the sinking fund formula. The chapter discusses the present value formula and present value of an ordinary annuity. A table that lists the amount of interest and amount of principal repayment in each periodic payment of an amortized loan is called an amortization schedule.

LINEAR PROGRAMMING (AN ALGEBRAIC APPROACH)

This chapter discusses the basic concepts of coordinate systems and graphs. For given two points P 1 ( x 1 , y 1 ) and P 2 ( x 2 , y 2 ), the slope of the line determined by P 1 and P 2 is given by the difference in the y coordinates divided by the corresponding difference of their x coordinates. Two distinct lines are parallel if they have equal slopes. The slope of a line measures the ratio of the change in y (increase or decrease) and the change in x when x increases. The change in the y direction is often called the rise. The change in the x direction is often called the run. The slope is a measure of the ratio of the rise to the run. If the slope is positive, then y increases when x increases and if the slope is negative, then y decreases when x increases. Vertical lines have no slope. A horizontal line has a slope of 0. Some specific exercises on coordinate systems and graphs are also discussed in the chapter.

CHAPTER 1 – SET THEORY

This chapter discusses matrices and linear systems. In solving a system of linear equations, the system of equations should be transformed into an equivalent system of equations that is easier to solve. A system of equations yields an equivalent system of equations if any two rows are interchanged; a row is multiplied by a nonzero constant; or a multiple of one row is added to another row. A matrix is in reduced row echelon form if: (1) a row is not made up entirely of zeros, then the leftmost nonzero number in the row is a 1; (2) there are any rows consisting entirely of zeros, they are all together at the bottom of the matrix; (3) in two successive rows, not consisting entirely of zeros, the first nonzero number in the lower row is to the right of the first nonzero number in the upper row; and (4) each column that contains the first nonzero number of some row has zeros everywhere else.

CHAPTER 9 – APPLICATIONS

This chapter describes the concepts of permutations and combinations. Given a set of distinct objects, an arrangement of these objects in a definite order without repetitions is called a permutation of the set. In general, a permutation of r objects selected from a set of n objects is called a permutation of the n objects taken r at a time. The number of permutations of a set with n distinct objects is n ( n −l)( n −2)….1. The number n ( n −l)( n −2)….1 is designated by the symbol n !. A subset of r objects selected from a set of n objects is called a combination of the n objects taken r at a time. In counting problems, it should be kept in mind that if order matters, permutations should be used and if order does not matter, combinations should be used. The multiplication principle tates that if there are k ways to make a decision D 1 and then n ways to make a decision D 2 , then there are kn ways to make the two decisions D 1 and D 2 .