1 Mathematical Preliminaries 1.1 Probability Theory Elements 1.1.1 Introduction Probability theory [326,357,365]is a part of theoretical and applied mathematics, which is engaged in establishing the rules governing random events-random games or experimental testing.The definitions,theorems and lemmas given below are necessary to understand the basic equations and computer implementation aspects used in the later numerical analyses presented in the book.They can also be used to calculate many of the closed-form equations applied frequently in applied sciences and engineering practice [19,37,150,201,202,253]. Definition The variations with n elements for k elements are k elements series where each number 1,2,...k corresponds to the single element from the initial set.The variations can differ in the elements or their order.The total number of all variations with n elements for k is described by the relation (n-k):=nn-D..(n-k+1) n! (1.1) k-times Example Let us consider the three-element set A(X,Y,Z).Two-element variations of this set are represented as V2=6:XY,YZ,XZ,YX,ZY,ZX. Definition Permutations with n elements are n-element series where each number 1,2,...,n corresponds to the single element from the initial n-element set.The difference between permutations is in the element order.The total number of all permutations with n different elements is given by the formula: Pn=V=1.2…n=nl (1.2) If among n elements X,Y,Z....there are identical elements,where X repeats a times,Y appears b times,while Z repeats c times etc.,then1 Mathematical Preliminaries 1.1 Probability Theory Elements 1.1.1 Introduction Probability theory [326,357,365] is a part of theoretical and applied mathematics, which is engaged in establishing the rules governing random events – random games or experimental testing. The definitions, theorems and lemmas given below are necessary to understand the basic equations and computer implementation aspects used in the later numerical analyses presented in the book. They can also be used to calculate many of the closed-form equations applied frequently in applied sciences and engineering practice [19,37,150,201,202,253]. Definition The variations with n elements for k elements are k elements series where each number 1,2,...,k corresponds to the single element from the initial set. The variations can differ in the elements or their order. The total number of all variations with n elements for k is described by the relation ( ) 14 2 44 4 3 44 k times k n n n n k n k n V − = − − + − = ( 1)...( 1) ! ! (1.1) Example Let us consider the three-element set A{X,Y,Z}. Two-element variations of this set are represented as 6 2 V3 = : XY, YZ, XZ, YX, ZY, ZX. Definition Permutations with n elements are n-element series where each number 1,2,...,n corresponds to the single element from the initial n-element set. The difference between permutations is in the element order. The total number of all permutations with n different elements is given by the formula: P V 1 2 ... n n! n n = n = ⋅ ⋅ ⋅ = (1.2) If among n elements X, Y, Z,... there are identical elements, where X repeats a times, Y appears b times, while Z repeats c times etc., then