Introduction to Mathematics & Statistics

SETS (1) A set is a well-specified collection of elements These elements can be finite or infinite Finite set: A = {1,2,3,4,5,6,7,8,9,10} Infinite set: B = {x | x>0} Membership to a set expressed as: a A Conversely: a A Empty set: A = { },also expressed as A =

SETS (2) Let A = {1,3,5,7}, B = {2,3,4,6,8} and C = {1,5}. Then it holds that: A C or C A, or else that C is a subset of A Let A = {1,2,3} and B = {4,5,6}. These two sets bear no common elements and are known as Disjoint Let A = {1,2,3,4,5} and B = {1,2,3,4,5}. Then these sets are called Equal

SET OPERATIONS Union: Let A = {2,3,6} and B = {4,3,5}; then their union is expressed as A U B = {2,3,4,5,6}. A U B = {x | x A or x B} Intersection: A B = {3}. A B = {x | x A and x B} Complement: Let A = {x | x 0) and Ā = {x | x < 0). If so, Ā is called the complementary set and A U Ā : R, where R is the set of all real numbers. Then it follows that: Ā = {x | x R and x A} Venn Diagrams

SETS OF NUMBERS Natural numbers: = {1, 2, 3, 4,.....} Integers:. = {0, 1, -1, 2, -2,.....} Rational numbers: = Real numbers: R R

THE VALUE OF MONEY (1): FUTURE VALUE Let us assume that one invests an amount of money at time zero (C 0 ) in an account for T years and that the going annual interest rate is r. Then, the value of this amount after the completion of T years shall equal: FV = C 0 (1+r) T where FV = Future Value However, the above assumes annual compounding of the interest, yet the latter could be compounded: Semiannually: FV = C 0 [1+(r/2)] 2T Quarterly: FV = C 0 [1+(r/4)] 4T Monthly: FV = C 0 [1+(r/12)] 12T Daily: FV = C 0 [1+(r/360)] 360T Continuously: FV = C 0 e rT

THE VALUE OF MONEY (2): FUTURE VALUE (B) In general, compounding an investment m times a year over many years provides wealth of FV = C 0 [1+(r/m)] mT

THE VALUE OF MONEY (2): PRESENT VALUE (A) Let us assume that an investor wishes to retain an amount C after T years of investment with the going annual interest rate equal to r. Then the amount of money he would have to invest now to obtain the amount C in the future is calculated as: PV = C T / (1 + r) T However, the above assumes annual compounding of the interest, yet the latter could be compounded: Semiannually: PV = C T/ [1+(r/2)] 2T Quarterly: PV = C T/ [1+(r/4)] 4T Monthly: PV = C T [1+(r/12)] 12T Daily: PV = C T [1+(r/360)] 360T Continuously: PV = C T e rT

THE VALUE OF MONEY (2): PRESENT VALUE (B) In case the expected cash flows are unequal in size, the present value is calculated as: PV = [C 1 / (1 + r) ] + [C 2 / (1 + r) ] 2 + [C 3 / (1 + r) ] 3 + [C 4 / (1 + r) ] 4 + [C 5 / (1 + r) ] 5 … [C T / (1 + r) ]

THE VALUE OF MONEY (3): SPECIAL CASES Perpetuity: constant stream of equal-sized cash-flows with no end PV = C / r Growing perpetuity PV = C / (r-g), where g is the rate of growth per period expressed as a percentage Annuity: level stream of regular payments that lasts for a fixed number of periods PV = C [ (1/r) – 1/(r(1+r) T ] Growing Annuity PV = C [ [1/(r-g)] – [1/(r-g)][(1+g)(1+r) T ]

EXPONENTIAL AND LOGARITHMIC FUNCTIONS x = a y log a x = y, a > 0 log a (x 1 x 2 ) = log(x 1 ) + log(x 2 ) log a (x 1 /x 2 ) = log(x 1 ) - log(x 2 ) log a a = 1 log a a x = x log a 1 = 0 log a x k = klog a x a x = e xlna (why??) lne = 1

LINEAR FUNCTIONS Y = aX + b These functions are called linear as their pictorial representation is given by a straight line The slope of a straight line is constant and can be calculated on the premises of the coordinates of two points on it. Let A (x 1,y 1 ) and B (x 2,y 2 ); then the slope is given by: m = (y 2- y 1 )/(x 2- x 1 ) If the slope and the coordinates of one point (x 0,y 0 ) are known, then the equation of a straight line can be given as: (y - y 0 )/(x - x 0 ) = m y = mx + (y 0 - m x 0 ) If two lines (let ε 1 and ε 2 ) are parallel to each other (ε 1 ε 2 ), then: m ε1 = m ε2 If two lines (let ε 1 and ε 2 ) are vertical to each other (ε 1 ε 2 ), then: m ε1 m ε2 =-1

NONLINEAR FUNCTIONS However, what happens when the function under question is not of a linear form? General form of quadratic equation: y = ax 2 + bx + c In this case the estimation of the slope (i.e. the rate of change over time) is facilitated through differential analysis

Differentiation Rules (1) Constant function rule then Power-function rule then Power-function rule generalized Sum-difference rule

Differentiation Rules (2) y = f(x) = lnx, then f(x) = 1/x, x>0 y = f(x) = e x, then f(x) = e x y = f(x) = a x, then f(x) = a x (lna)

Differentiation Rules (3) Product rule Quotion rule Chain rule

Maxima and minima First-derivative test f(x 0 ) = 0 A relative maximum if f(x 0 ) changes from positive to negative from the left of point x 0. A relative minimum if f(x 0 ) changes from negative to positive from the left of point x 0. Neither a relative maximum nor a relative maximum if f(x 0 ) has the same sign on both left and right of point x 0. Second derivative f(x), f(x)>0 means the slope tends to increase, f(x)<0 means the slope tends to decrease

Maxima and minima A second derivative test: If f(x 0 ) = 0 then at x 0 f(x 0 ) will be a)A relative maximum if f(x 0 ) < 0 (concavity) b)A relative minimum if f(x 0 ) > 0 (convexity)

Partial derivatives Ex and geometric interpretation Jacobian determinants

INTEGRATIONAL ANALYSIS The inverse process of differentiation Example: assume we know the growth rate of a population and we wish to establish its size at time t For any given function (let, f) differentiable in a domain (let Δ), the following holds: f(x)dx = f(x) + c, x Δ

Basic rules of integration (1) The power rule The exponentiation rule The logarithmic rule Augmented Exponentiation Rule Augmented Logarithmic Rule

Basic rules of integration (2) The integral of a sum The integral of a multiple Integration by substitution

Basic rules of integration (3) Integration through factorization

MATRIX ALGEBRA System of m linear equation and n variables Matrices as Arrays

DEFINITIONS Matrix: a two-dimensional vector (m x n), with m rows and n columns. Let matrices A (m a x n a ) and B (m b x n b ). A = B, if and only if: m a = m b, n a = n b and all their elements are equal. Square matrix Row-vector (1 x n) Column-vector (m x 1) Matrix-diagonal Triangular-up/-down matrix Identity matrix Null matrix

ADDING/SUBTRACTING MATRICES Possible only if the matrices dimensions are identical A [a ij ], B [b ij ] A ± B = [a ij ± b ij ] A + B = B + A A + (B + C) = (A + B) + C A + Ο = O + A = A A + (-A) = (-A) + A = O A – B = A + (-B) X + B = A X = A - B

MATRIX-MULTIPLICATION: MULTIPLYING MATRICES WITH NUMBERS Let A = [a ij ], B = [b ij ] k x A = [k x a ij ], k R (k + z) x A = k x A + z x A, k, z R k (A + B) = k x A + k x B, k R (k x z) A = k x (z x A), k, z R k x A = O k = 0, or A = O

MATRIX-MULTIPLICATION: MULTIPLYING MATRICES WITH MATRICES Consider the matrices A = [a ik ], B [b kj ] Let A m x n and B n x r We define as AxB = [c ij ], where: c ij = a i1 b 1j + a i2 b 2j + a i3 b 3j + …+ a in b nj Each element is the sum of the products of all n-elements of As i-th row multiplied with their corresponding n-elements of Bs j-th column

MULTIPLYING MATRICES WITH MATRICES: ESSENTIAL PROPERTIES As long as multiplication is possible: A(BC) = (AB)C A(B+C) = AB + AC (kA)(zB) = (kz)(AB) AO = OA = O AI = IA = A If A m x m, then one can estimate the following products: AA = A 2, AAA = A 3 et al A 1 = A A p A q = A p+q (A p ) q = A pq (kA) p = k p A p, k R If AB = BA, then the following hold: (A±B) 2 = A 2 ± 2AB + B 2 (A ±B) 3 = A 3 ±3A 2 B + 3AB 2 ±B 3 A 2 – B 2 = (A – B)(A + B) A 3 – B 3 = (A-B)(A 2 + AB +B 2 )

INVERTIBLE MATRICES AB = BA = I The above holds if and only if A, B are m x m A = invertible B = inverse Each matrix has a unique inverse: AA -1 = A -1 A = I If A, b, mxm, then AB = I if and only if BA = I As a result, to prove that a matrix is invertible, all we need to prove is one the following: Either AB = I or BA = I If a matrix A is invertible, then it follows that: AX = B X = A -1 B XA = B X = B A -1

A final note… Let A, B with AB = O It may well be the case that neither of these matrices is a null one. However, if the above holds and one of the two is invertible, then the other is a null one

Determinants The determinant of a square matrix A, denoted by |A| is a uniquely defined scalar associated with that matrix. Defined only for square matrices. For a 2×2 matrix The determinant is equal to:

Determinants Evaluating a third order determinant Its determinant has the value

Solving a nxn linear system using Cramers method Let the nxn linear system AX = B If |A| 0, then the system has only one solution (x 1, x 2,…, x n ) with X 1 = Dx 1 /D,x 2 = Dx 2 /D, …,x n = Dx n /D Where D is the determinant |A| of the unknown variablescoefficients and Dx i, i = 1,2,3, … n is the determinant resulting from substituting the ith column of the X i coefficients with the constants column. If |A| = 0, then the system either has no solutions or has infinite solutions