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Appendix 7. Elementary mathematical and statistical concepts

Logarithm : Logarithm of a number N to the base a is the number x to which the base must be raised to equate with the original number N. i.e., if loga N = x, then ax = N. The number N is called antilogarithm of x. Logarithm to the base 10 is called the common logarithm (indicated by log) and to the base e, a mathematical constant, is called natural logarithm (indicated ln).

Factorial n : Factorial n, denoted by n!, is defined as n! = n(n-1)(n-2)1. Thus 5! = 5.4.3.2.1 = 120. It is convenient to define 0! = 1.

Combinations : A combination of n different objects taken r at a time is a selection of r out of the n objects with no attention given to the order of arrangement. The number of combinations of n objects taken r at a time is denoted by and is given by

=

For example, the number of combinations of the letters a, b, c taken two at a time is These are ab, ac, bc. Note that ab is the same combination as ba, but not the same permutation.

Mathematical expectation : If X denotes a discrete random variable which can assume the values X1, X2, , Xk with respective probabilities p1, p2, , pk where p1+ p2+ + pk = 1, the mathematical expectation of X or simply the expectation of X, denoted by E(X), is defined as

E(X) = p1X1 + p2X2 + + pkXk .

In the case of continuous variables, the definition of expectation is as follows. Let g(X) be a function of a continuous random variable X, and let f(x) be the probability density function of X. The mathematical expectation of g(x) is then represented as

where R represents the range (sample space) of X, provided the integral converges absolutely.

Matrix : A matrix is a rectangular array of numbers arranged in rows and columns. The rows are of equal length as are the columns. If aij denote the element in the ith row and jth column of a matrix A with r rows and c columns, then A can be represented as,

Ar x c = A = {aij}

A simple example of a 2 x 3 matrix is A 2 x 3=

A matrix containing a single column is called a column vector. Similarly a matrix that is just a row is a row vector. For example, x = is a column vector. y = is a row vector. A single number such as 2, 4, -6 is called a scalar.

The sum of two matrices A = {aij} and B = {bij} is defined as C ={cij} = {aij+ bij}. For example, if,

A = and B = , then C =

The product of two matrices is defined as Cr x s = Ar x c Bc x s where the ijth element of C is given by cij = . For example, if,

A = and B = , then C =

For further details and examples from biology, the reader if referred to Searle (1966).

 

 

 

 

 

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