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
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