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:

2 2( ) / 21( )

2yf y e

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,

2var( ) var( )cy d c y

2var( ) [ ( )]y E y E y

2 2var( ) var( ) var( )cy dz c y d z

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

cov( , ) [( ( ))( ( )]y z E y E y z E z

cov( , ) cov( , )c dy e fz df y z

Characteristics of probability distributions

• If a random sample of size T: y1, y2, ..., yn 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

y

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 = 2,7128....

• ln(xy) = ln(x) + ln(y)• ln (x/y) = ln(x) – ln(y)• ln(yc) = 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 xn, dy/dx = cnxn-1

• E.g.: y = 4x3, dy/dx = 12 x2

•

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(x3+2x-1))=(3x2+2)/(x3+2x-1)

Differential calculus• Rules:

• The derivative of ex = ex

• The derivative of the ef(x)= f’(x)ef(x)

• If y = f(x1, x2, ..., xn), the differentiation of y wrt only one variable is the partial differentiation: – E.g. 3 4 2

1 1 2 2

21

1

3 4 2 2

9 4

y x x x x

yx

x

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

MatricesA 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.

11 12 1

21 22 2

1 2

...

...A

...

n

n

m m mn

a a a

a a a

a a a

( 1, ; 1, )ija i m j n

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.:

3 42

7 9

3 61

2 4

rank

rank

Matrix Operations

• Equality: A = B if and only if A and B have the same size and aij = bij i, j.

• Addition of matrices: A+B= C if and only if A and B have the same size and aij + bij = cij i, j.

2 4 1 2 1 6

3 5 4 1 7 6

Matrix operations

• Multiplication of a scalar by a matrix:

k.A = k.[aij], 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.BB.A

1

n

ij ik kjk

c a b

2 4 1 2 2 ( 1) 4 4 2 2 4 1 14 8

3 5 4 1 3 ( 1) 5 4 3 2 5 1 17 11

Transpose of a matrix

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

11 12 1 11 21 1

21 22 2 12 22 2

1 2 1 2

... ...

... ...A ; A '

... ...

n m

n m

m m mn n n mn

a a a a a a

a a a a a a

a a a a a a

Properties of transpose matrices

• (A+B)+C=A+(B+C)

• (A.B).C=A(B.C)

i. (A')'=A

ii. (A+B)'=A'+B'

iii. (A.B)'=B'.A'

iv. If A is square and if A=A', then A is symmetrical.

Square matrices :

• Identity matrix I:

Note: A.I = I.A = A, where A has the same size as I.

1 0 0 0

0 1 0 0I

0 0 1 0

0 0 0 1

Square matrices :

• Diagonal matrix:

1

2

0 ... 0

0 ... 0

0 0 ... n

Square matrices:

• Scalar matrix = diagonal matrix, when

n .

• Zero matrix: A + 0 = A; A x0 = 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)

1

(A)

( A) ( (A))

n

iii

tr a

tr c c tr

Determinants

• matrix 2 x 2:

3 13 2 2 1 6 2 4

2 2

Determinants • matrix 3 x 3:

2 3 2

1 1 2

3 2 2

1 2 1 2 1 12 3 2

2 2 3 2 3 2

2(2 4) 3(2 6) 2(2 3)

4 12 2 6

Determinants

• Matrix 3 x 3: Kramer’s rule

2 3 2 2 3 2 2 3

1 1 2 1 1 2 1 1

3 2 2 3 2 2 3 2

2 1 2 3 2 3 2 1 2 2 1 3 2 2 2 3 1 2

4 18 4 6 8 6 6

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 aij 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.

i. Adjunct matrix is the transpose of the cofactor matrix, i.e. adj A = C’.

ii. Inverse matrix: 1 1A A

Aadj

11 12 11 12

21 22 21 22

11 11 12 12

22 12

21 11

1

11 22 12 21 11 22 12 2

11 22 11 22

12 21 12 21A

21 12 21 12

22 11 22 11

A =

adj A = C'=

1A

m a c a

m a c aa a a aC

m a c aa a a a

m a c c

a c a c

a a

a a

adjA

a a a c a a a c

22 12

1 21 11

a a

a a

Example2 x 2 matrix :

-1

4 3A A 4 3 2 3 12 6 6

2 3

3 2 3 -3C A = C' =

3 4 -2 4

3 -3 0,5 0,51 1A A=

-2 4 0,33 0,66A 6

adj

adj

Example• 3 x 3 matrix :

1 2 1 2 1 1

2 2 3 2 3 22 3 2 2 4 1

3 2 2 2 2 31 1 2 det 6 2 2 5

2 2 3 2 3 23 2 2 4 2 1

3 2 2 2 2 3

1 2 1 2 1 1

2 2 4

6 6 64 2 2

6 6 61 5 1

6 6 6

A A cofactor matrix

Inverse

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 = Ipc where Ip is an identity matrix, and hence

• ( – Ip) c = 0

The eigenvalues of a matrix

• Since c 0 the matrix ( – Ip) must be singular (zero determinant)

– Ip = 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

p

2

5 1 5 1 1 0 - I 0

2 4 2 4 0 1

5 1(5 )(4 ) 2 9 18 0

2 4

6 3