A Review of Some Fundamental Mathematical and Statistical Concepts UnB Mestrado em Ciências Contábeis Prof. Otávio Medeiros, MSc, PhD

Characteristics of probability distributions Random variable: can take on any value from a given set Most commonly used distribution: normal or Gaussian Normal probability density function:

Characteristics of probability distributions The mean of a random variable is known as its expected value E(y). Properties of expected values: –E(c) = c c = constant –E(cy) = c E(y) –E(cy + d) = c E(y) + d d = constant –If y and z are independent, E(yz) = E(y) E(z)

Characteristics of probability distributions The variance of a random variable is var(y). Properties of var operator: var(c) = 0 –. –If y and z are independent,

Characteristics of probability distributions The covariance between 2 random variables is cov(y,z). Properties of cov operator: –If y and z are independent, cov(y,z) = 0 –If c, d, e, and f are constants

Characteristics of probability distributions If a random sample of size T: y 1, y 2,..., y n is drawn from a normally distributed population with mean  and variance  , the sample mean is also normally distributed with mean  and variance   /T. Central limit theorem: sampling distribution of the mean of any random sample tends to the normal distribution with mean  as sample size 

Properties of logarithms Logs can have any base, but in finance and economics the neperian or natural log is more usual. Its base is the number e = 2, ln(xy) = ln(x) + ln(y) ln (x/y) = ln(x) – ln(y) ln(y c ) = c ln(y) ln(1) = 0 ln(1/y) = ln(1) – ln(y) = – ln(y)

Differential calculus The effect of the rate of change of one variable on the rate of change of another is measured by the derivative. If y = f(x) the derivative of y w.r.t x is dy/dx or f’(x) measures the instantaneous rate of change of y wrt x

Differential calculus Rules: The derivative of a constant is zero If y = 10, dy/dx = 0 If y = 3x + 2, dy/dx = 3 If y = c x n, dy/dx = cnx n-1 E.g.: y = 4x 3, dy/dx = 12 x 2

Differential calculus Rules: The derivative of a sum is equal to the sum of the derivatives of the individual parts: –E.g. If y = f(x)+g(x), dy/dx = f’(x)+g’(x) The derivative of the log of x is given by 1/x –d(log(x))/dx = 1/x The derivative of the log of a function: – d(log(f(x)))/dx=f’(x)/f(x) –E.g. d(log(x 3 +2x-1))=(3x 2 +2)/(x 3 +2x-1)

Differential calculus Rules: The derivative of e x = e x The derivative of the e f(x) = f’(x)e f(x) If y = f(x 1, x 2,..., x n ), the differentiation of y wrt only one variable is the partial differentiation: –E.g.

Differential calculus The maximum or minimum of a function wrt a variable can be found setting the 1st derivative f’(x) equal to zero. Second order condition: –If f”(x)>0  minimum –If f”(x)<0  maximum

Matrices A Matrix is a collection or array of numbers Size of a matrix is given by number of rows and columns R x C If a matrix has only one row, it is a row vector If a matrix has only one column, it is a column vector If R = C the matrix is a square matrix

Definitions Matrix is a rectangular array of real numbers with R rows and C columns. are matrix elements.

Definitions Dimension of a matrix: R x C. Matrix 1 x 1 is a scalar. Matrix R x 1 is a column vector. Matrix 1 x C is a row vector. If R = C, the matrix is square. Sum of elements of leading diagonal = trace. Diagonal matrix : square matrix with all elements off the leading diagonal equal to zero. Identity matrix: diagonal matrix with all elements in the leading diagonal equal to one. Zero matrix: all elements are zero.

Definitions Rank of a matrix: is given by the maximum number of linearly independent rows or columns contained in the matrix, e.g.:

Matrix Operations Equality: A = B if and only if A and B have the same size and a ij = b ij  i, j. Addition of matrices: A+B= C if and only if A and B have the same size and a ij + b ij = c ij  i, j.

Matrix operations Multiplication of a scalar by a matrix: k.A = k.[a ij ], i.e. every element of the matrix is multiplied by the scalar.

Matrix operations Multiplication of matrices: if A is m x n and B is n x p, then the product of the 2 matrices is A.B = C, where C is a m x p matrix with elements: Example: Note: A.B  B.A

Transpose of a matrix matrix transpose: if A is m x n, then the transpose of A is n x m, i.e.:

Properties of transpose matrices (A+B)+C=A+(B+C) (A.B).C=A(B.C)

Square matrices : Identity matrix I: Note: A.I = I.A = A, where A has the same size as I.

Square matrices : Diagonal matrix:

Square matrices: Scalar matrix = diagonal matrix, when     n. Zero matrix: A + 0 = A; A x  0 = 0.

Trace: If A is m x n and B is n x m, then AB and BA are square matrices and tr(AB) = tr (BA)

Determinants matrix 2 x 2:

Determinants matrix 3 x 3:

Determinants Matrix 3 x 3: Kramer’s rule

Inverse matrix The inverse of a square matrix A, named A -1, is the matrix which pre or post multiplied by A gives the identity matrix. B = A -1 if and only if BA = AB = I Matrix A has an inverse if and only if det A  0 (i.e. A is non singular). (A.B) -1 = B -1.A -1 (A -1 )’=(A’) -1  if A é symmetrical and non singular, then A -1 is symmetrical. If det A  0 and A is a square matrix of size n, then A has rank n.

Steps for finding an inverse matrix Calculation of the determinant: Kramer’s rule or cofactor matrix. Minor of the element a ij is the determinant of the submatrix obtained after exclusion of the i-th row and j-th column. Cofactor is the minor multiplied by (-1) i+j,

Steps for finding an inverse matrix Laplace expansion: take any row or column and get the determinant by multiplying the products of each element of row or columns by its respective cofactor. Cofactor matrix: matrix where each element is substituted by its cofactor.

Example 2 x 2 matrix :

Example 3 x 3 matrix :

The eigenvalues of a matrix Let  be a p x p square matrix and let c denote a p x 1 non-zero vector, and let denote a set of scalars. is called a characteristic root of the matrix  if it is possible to write:  c = I p c where I p is an identity matrix, and hence (  – I p ) c = 0

The eigenvalues of a matrix Since c  0 the matrix (  – I p ) must be singular (zero determinant)  – I p  = 0

The eigenvalues of a matrix Example: Characteristic roots = eigenvalues The sum of eigenvalues = trace of the matrix The product of the eigenvalues = determinant The number of non-zero eigenvalues = rank