When we multiply or divide an inequality by a negative integer, the sign of the inequality will be reversed (changed). x>y, x*z<y*z if z<0

factorization -(x – 2)(x + 2) is a factorization of the polynomial x^2 – 4

greatest common factor or divisor (GCF or GCD)

• Its direct application is not often observed in daily life and is associated with high school education. Arithmetic is generally applicable in real life and associated with elementary education.
• It uses numbers, variables, and general rules or formulae to solve problems. Not just 4 operatios like in Arithmetic.

• set - множество.C = {0,1,2}
• Cardinality [kɑːdɪˈnælɪtɪ] (мощность) - |B| = количество эллементов (elements or members)
• x ∈ B - x belongs to B
• y ∉ B - y does not belong to B
• A ⊆ B - A is a subset of B or A is contained in B - every member of set A is also a member of set B
• proper (or strict) subset - f A is a subset of B, but A is not equal to B
• ∅ - empty set - ∅ ⊆ A.
• A ⊆ U - universal set
• A ∪ B - Union - together
• A ∩ B - Intersections [ɪntəˈsekʃn] A ∩ ∅ = ∅ - пересечение - общее
• {1, 2, 3, 4} \ {1, 3} = {2, 4}. - Set difference
• A' = U \ A is called the absolute complement or simply complement of A
• {1, 2} \ {1, 2} = ∅.
• A ∩ A' = ∅. - A' or A^c = Complement
• Cartesian product - A x B - {1, 2} × {red, white, green} = {(1, red), (1, white), (1, green), (2, red), (2, white), (2, green)}.
• |A × B| = |A| × |B| = умножение
• union ∪ and intersection ∩: Commutative, Associative, Distributive
• absorption laws:
  • A∪(A∩B)=A
  • A∩(Α∪B)=A
• A∩B=A\(A\B)
• algebra of inclusion ⊆:
  • reflexivity A⊆A - ∀x∈A:x⊆x
  • antisymmetry A⊆B and B⊆A only if A=B
  • transitivity. If A⊆B and B⊆C, then A⊆C
• equivalent:
  • A⊆B
  • A∩B=A
  • A∪B=B
  • A\B=0
  • B'⊆A'
• complement lows:
  • (A∪B)' = A'∩B'
  • (A∩B)' = A'∪B'
  • (A')'= A
  • 0' = U
  • U' = 0
• Transitive closure - binary relation R on a set X smallest relation on X that contains R and is transitive.
  • if X is a set of airports. xRy means "there is a direct flight from airport x to airport y. xR+y means "it is possible to fly from x to y in one or more flights
  • intersection of two R+ is transitive
  • union need not to be transitive.

Для илюстрации используются диагрмы эйлера Euler diagram.

is a branch of mathematics concerned with relationships between angles and ratios of lengths.

Graph - pair (V,E), where V={a, b, c} finite set of vertix and E is a finite set of edges E={ab, ac}

edge [ɛdʒ]

ребро - joining a pair of nodes

vertex [ˈvɜːtɛks] vertices [ˈvɜːtɪˌsiːz] node [nəʊd]

вершиныx

arc [ɑːk]

дуга is a directed line (a pair of ordered vertices).

Incident [ˈɪnsɪdənt]

смежный edge

adjacent [əˈdʒeɪsənt]

примыкающий vertex

successor [səkˈsɛsə]

приемник vertex

walk [wɔːk]

тропа

circuit [ˈsɜːkɪt]

контур

cycle [ˈsaɪkəl]

цикл

girth [ɡɜːθ]

обхват

circumference [səˈkʌmfərəns]

длина окружности

Hypergraph [haɪpəˈgræf]

edge can join any number of vertices H=(X,E) X - nodes or vertices, E - non-empty subsets of X

Main properties:

• radius of a connected graph r(G) - minimum eccentricity from all the vertices
• diameter d(G) - maximum eccentricity from all to all other vertices
• girth [ɡɜːθ] - number of edges in the shortest cycle of ‘G’

• Incident edges are edges which share a vertex. A edge and vertex are incident if the edge connects the vertex to another.
• A loop is an edge or arc that joins a vertex to itself.
• A vertex, sometimes called a node, is a point or circle. It is the fundamental unit from which graphs are made.
• Adjacent vertices are vertices which are connected by an edge.
• The degree of a vertex is simply the number of edges that connect to that vertex. Loops count twice.
• A predecessor is the node (vertex) before a given vertex on a path.
• A successor is the node (vertex) following a given vertex on a path.
• A walk is a series of vertices and edges. A u-v walk would be a walk beginning at u and ending at v.
• A closed walk is a walk from a vertex back to itself; a series of vertices and edges which begins and ends at the same place. May have repeated vertices or/and edges.
• A circuit is a closed walk with every edge distinct.
• A cycle is a closed walk with no repeated vertices (except that the first and last vertices are the same).
• A path is a walk where no repeated vertices. A u-v path is a path beginning at u and ending at v.
• isolated vertex has no edges.
• indegree = deg-(V), outdegree = deg+(V)
• pendent vertex = vertex deg(v) = 1.
• degree sequence for V={b, a, c, d} and E={ba, bc} will be {2, 1, 1, 0}
• distance between 2 vertices - number of deges in a shortest path between U and V = d(U, V)
• eccentricity of a vertex [ˌɛksɛnˈtrɪsɪtɪ] e(V) - max distance to all other.
• central point e(V) =r(V)
• centre [ˈsɛntə] set of all central points
• circumference [səˈkʌmfərəns] - number of edges in the longest cycle of ‘G’
• Connected undirected graph. 1) there is a path between every pair of vertices
• Connected component of undirected graph. A subgraph in which any two vertices are connected to each other by paths
• Incidence matrix columns - vertices, rows - edges
• Degree matrix - diagonal matrix
• Laplacian matrix

• edge contraction [kənˈtrækʃən] is an operation which removes an edge from a graph while simultaneously merging the two vertices that it previously joined
• graph traversal [ˈtrævɜːs] is an exploration of a graph in which the vertices are visited or updated one by one.
• A Hamiltonian cycle is a closed loop where every node is visited exactly once.

• Circuit Rank of G (cyclomatic number, cycle rank, or nullity) minimum number of edges that must be removed from the undirected connectedgraph to break all its cycles, making it into a tree or forest. r = m - n + c where m= edges, n= vertices, c = number of connected components.
• Laplacian matrix (admittance matrix, Kirchhoff matrix or discrete Laplacian) -

• null graph no edges.
• trivial graph one vertix, no edges.
• A multigraph is a graph with multiple edges(parallel edges) with or without loops.
• simple grapth no parallel edges, no loops. deg(V) <= n-1 ∀ v ∈ G
• directed and non-directed [dɪˈrɛktɪd]. edge has direction ab=ba or not.
• Connected graph - undirected graph when there is a path between every pair of vertices.
• Bipartite Graph - двудольный - with vertex partition V = {V1, V2} if every edge of E joins a vertex in V1 to a vertex in V2
  • Complete Bipartite Graph - bipartite graph - if every vertex in V1 is connected to every vertex of V2.
• complete graph - simple undirected graph - every pair of distinct vertices is connected by a unique edge

• tree - any two vertices are connected by exactly one path or a connected acyclic undirected graph.
  • if V = n, E = n-1.
• tree - disconnected acyclic graph
• H Spanning Trees of G - основное дерево. Where G is connected graph. H is a tree and has all vertices of G.

• An acyclic directed graph [eɪˈsaɪklɪk] is a finite directed graph which has no directed cycles.
• A condensation [ˌkɒndɛnˈseɪʃən] of a multigraph is the graph that results when you delete any multiple edges, leaving just one edge between any two points.
• If a graph has a path between every pair of vertices (there is no vertex not connected with an edge), the graph is called a connected graph. If a graph G’ can be constructed from a graph G by repeated edge contractions or deletions, the graph G’ is a graph minor of G. An inverted graph G’ of G is a graph with the same vertices but none of the same edges; two vertices in G’ are adjacent if and only if they were not adjacent in G. A multigraph is a graph without loops, but which may have multiple edges. A null graph is a graph with no edges. It may have one or more vertices. An oriented graph is a directed graph that doesn’t have any symmetric pairs of directed edges. A simple graph is a graph that doesn’t have any loops or multiple edges. No multiple edges means that no two edges have the same endpoints. A subgraph is a graph whose vertices and edges are included in the vertices and edges of another graph (the supergraph). A symmetric graph is a directed graph D where, for every arc (x,y), the inverted arc (y,x) is also in D. A trivial graph is a graph with only one vertex. An undirected graph is a graph where none of the edges have direction; the pairs of vertices that make up each edge are unordered.

• https://www.statisticshowto.datasciencecentral.com/graph-theory/
• https://www.tutorialspoint.com/graph_theory/graph_theory_quick_guide.htm

• https://ru.wikipedia.org/wiki/%D0%93%D0%BB%D0%BE%D1%81%D1%81%D0%B0%D1%80%D0%B8%D0%B9_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D0%B8_%D0%B3%D1%80%D1%83%D0%BF%D0%BF#%D0%9A
• https://en.wikipedia.org/wiki/List_of_group_theory_topics
• group - set with binary operation • (group law of G) to form a third element in such a way that four group axioms are satisfied:
  • closure For all a, b in G, the result of the operation, a • b, is also in G.
  • associativity For all a, b and c in G, (a • b) • c = a • (b • c).
  • identity There exists an element e such that, for every element a the equation e • a = a • e = a holds. e - identity element.
  • invertibility For each a exists an element b (a^-1 or -a) such that a • b = b • a = e. e - identity element.

Абелевые группы: Convention Operation Identity Powers Inverse Addition x + y 0 nx −x Multiplication x ⋅ y or xy e or 1 x^n x^−1

• center of a group - set of elements that commute with every element of G. Z(G) = {z ∈ G ∣ ∀g ∈ G, zg = gz}
• Abelian group - коммутативная -obey the axiom of commutativity[kəˈmjuːtətɪvɪtɪ] .
• Group homomorphisms - are functions that preserve group structure. G → H between two groups (G, •) and (H, ∗) where a(g • k) = a(g) ∗ a(k) - (∀g,k∈G
• Isomorphic groups [aɪsəˈmɔːfɪk] - if there exist group homomorphisms a: G → H and b: H → G such that a(b(h)) = h and b(a(g)) = g for any g in G and h in H
• Identity element - Нейтральный элемент 1: a · 1 = a and 1 · a = a
• partial binary operation - f:X->Y not forcing f to map every element of X to an element of Y. Ex. division of real numbers - a/0 is not defined for any real a. However, both in universal algebra and model theory binary operations considered are defined on all of S × S.
• Magma - базовая математическая структура. Set with closed binary operation. magma or closure axiom: For all a, b in M, the result of the operation a • b is also in M. If partial operation, then a partial magma.
• Semigroup - set S with a associative binary operation
• monoid or groupoid [məˈnɔɪd] - полугруппа с нейтральным элементом
• Ring - set R with two binary operations + and ring axioms:
  1. R is an abelian group under addition
     • associative
     • commutative
     • additive identity - There is an element 0 in R such that a + 0 = a for all a in R
     • additive inverse - For each a in R there exists −a in R such that a + (−a) = 0
  2. R is a monoid under multiplication
     • associative
     • multiplicative identity
  3. Multiplication is distributive with respect to addition
     • left distributivity a ⋅ (b + c) = (a · b) + (a · c) for all a, b, c in R
     • right distributivity (b + c) · a = (b · a) + (c · a) for all a, b, c in R
• Ring properties:
  • The additive identity, the additive inverse of each element, and the multiplicative identity are unique.
• Field -
• Difference between groups and fields - Groups model symmetries
• Matrix ring -
• Matrix [ˈmeɪtrɪks] - rectangular array of numbers, symbols, or expressions, arranged in rows and columns
• dimensions - 2x3 "two and three"
  • 1 2 3
  • 1 2 3

Uniqueness of inverses - b = b • e as e is the identity element = b • (a • c) Because c is an inverse of a, so e = a • c = (b • a) • c by associativity, which allows rearranging the parentheses = e • c since b is an inverse of a, i.e., b • a = e = c for e is the identity element there is only one inverse element of a

Uniqueness of identity element G is a group with two identity elements e and f. Then e = e • f = f, hence e and f are equal.

see 14.2.1 Это метрика расстояния для Real coordinate space. Simple case: Cartesian coordinates of the points of a Euclidean space.

Используется для задания Euclidean space.

скалярное произведение это функция

• для любыъ векторов u,v,w и и любых чисел a,b: (a*u+b*v,w) = a(u,w)+b(v,w)
• (u,v) = (v,u)
• (u,u) >= 0

Пример такого пространства, это набор вечественных чисал (x1,x2,x3), где скалярное произведение:

• (x,y) = ∑xi*yi = x1*y1+x2*y2+x3*y3

Определение скалярного произведения достаточно, чтобы ввести понятия

• длина |u|=sqrt((u,u))
• угол = arccos((x,y)/(|x|*|y|))

set whose elements are vectors, may be added together and multiplied ("scaled") by numbers called scalars. (generalize Euclidean vectors)

Векторы коллинеарны если они одно их двух

• независимы 9.1
• их векторное произведение коллинеарных векторов равно 0

Affine space is a set A with a vector space, and a transitive and free action of the additive group of verctor space on set A.

• The elements of the affine space A are called points.
• The vector space is said to be associated to the affine space, and its elements are called vectors.
• action generally denoted as an addition.
  • a+0=a
  • (a+u)+w = a+(u+w), where a ∈ set A and u,w in vector space A
  • for any m and n in set A, there is only one v ∈ vector space A: N=M+v

function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin:

• subadditivity/ triangle inequality: p(x+y) <=p(x) + p(y)
• Absolute homogeneity: p(s*x) = |s|*p(x), for all x ∈ X and all scalars s.
• Positive definiteness/Point-separatin: if p(x) = 0, then x = 0

linear comany form of x and y is ax + by, where a and b is constants

• V1: a1*x+b1*y or (a1,b1)
• V2: a2*x+b2*y or (a2,b2)
• V1+V2 = a1*x+b1*y + a2*x+b2*y = (a1+a2)*x + (b1+b2)*y
• or a*V1+b*V2
• нетривиальной, если хотя бы один из её коэффициентов отличен от нуля.
• барицентрической, если сумма её коэффициентов равна 1[4],
• выпуклой, если сумма её коэффициентов равна 1 и все коэффициенты неотрицательны,
• сбалансированной, если сумма её коэффициентов равна 0.

Linear independence - if a1*v1 + a2*v2 = 0 and a1 != 0 or a2 !=0 - linear dependent

• for vectors - they are not collinear if they are independent
• for Линейные системы уравнений - имеет однозначное решение тогда, когда столбцы её основной матрицы являются линейно независимыми.
• matrixs: Ранг матрицы равен максимальному числу её линейно независимых строк или столбцов.

determinant of a coefficient matrix (let it be |A| )

• == 0 => no solutions or infinite solutions.
• != => unique solution

• y = (i=1-3)∑ci*xi = c1*x1+c2*x2+c3*x3 == y = ci*x^i
• uk=ai*bk^i == uk = (i=1-n)∑ai*bk^i где n - размерность a, b
• e'i = Si^j*ej - суммирование по j

• https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D0%B2%D0%B0%D1%80%D0%B8%D0%B0%D0%BD%D1%82%D0%BD%D0%BE%D1%81%D1%82%D1%8C_%D0%B8_%D0%BA%D0%BE%D0%BD%D1%82%D1%80%D0%B0%D0%B2%D0%B0%D1%80%D0%B8%D0%B0%D0%BD%D1%82%D0%BD%D0%BE%D1%81%D1%82%D1%8C_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)

• Addition
  • require equal dimensions
  • (A+B)ij = Aij+Bij
• Scalar multiplication - (cA)ij= c*Aij
• Transposition - A m-by-n to A^tr or A^T n-by-m (A^T)ij = Aji - переворачивает относительно диагонали
  • A*A^T = symmetric matrices

• Matrix multiplication - используются для преобразования векторов
  • require A n columns equal to B m rows
  • 2*1000 + 3*100 + 4*10 = 2340
  • associativity (AB)C = A(BC)
  • left and right distributivity (A + B)C = AC + BC as well as C(A + B) = CA + CB
  • not commutative AB =! BA

Определитель (детерминант)

• require квадратная
• Δ = det A
• for 2x2 Δ = a11a22 - a12a21

Элементарные преобразования не изменяют множество решений системы линейных алгебраических уравнений, которую представляет эта матрица.

Элементарная матрица - если умножение на неё произвольной матрицы В приводит к элементарным преобразованиям строк в матрице В.

A*A^-1 = A^-1*A = E - единичная матрица

квадратная матрица, элементы главной диагонали которой равны единице поля, а остальные равны нулю.

Свойства: AE=EA=A A^0 = E A*A-1

это коэффициенты при x^k в разложении бинома Ньютона (1+x)^n

• (1+x)^n = ∑Cnk*x^k
• обозначается Cnk или вертикальные скобочки (n k)
• Cnk = n!/(k!(n-k)!) для 0<=k<=n. Если k>n Cnk=0
• факториал n! = 1*2*..*n
• (n-1)!=n!/n => 0!=1
• 0!=1, 1!=1, 2!=2, 3!=6, 4!=24, 5!=120
• файкториал 4!=24 - это количество перестановок для множества {A,B,C,D}

Probability distribution - function whose value at any given sample (or point - outcome) can be interpreted as providing a relative likelihood (or probability) that the value of the random variable would equal that sample.

P:A->R [0,1]

Методы описания (распределения случайной величины):

• F(x) функции распределения - вероятностью того, что значения случайной величины меньше или равное вещественного числа x - интервал [a, b) - вероятность F(b)-F(a)
• f(x) - плотности вероятности density
• характеристической функции
• probability mass function (PMF)
• cumulative distribution function

1. Probability quantifies as a number between 0 and 1
2. P(Ω) = 1 - probability that at least one of the elementary events
3. P(∪Ei) = ∑P(Ei) Where E - events. Any countable sequence of disjoint sets (synonymous with mutually exclusive events) satisfies. - взаимоисключающие события

Note:

• P(E) - probability of some event
• Let (Ω, F, P) be a probability space -
• Ω - set of all possible outcomes
• F - set of events - each event is a set containing zero or more outcomes - или Алгебра событий
• elementary event or simple event - is an event which contains only a single outcome
• P(not A) = 1 - P(A)

Example: single coin-toss, and assume that the coin will either land heads (H) or tails (T) (but not both). No assumption is made as to whether the coin is fair.

• Ω = {H,T}
• F={∅,{H},{T},{H,T}}
• P(∅)=0 - Вероятность невыподания = 0
• P({H,T})= 1 = P({H}) + P({T})

Consequences: P(A∪B)=P(A) + P(B) - P(A∩B) becouse of:

• P(A∪B) = P(A) + P(B\A) (by Axiom 3)
• P(A∪B) = P(A) + P(B\(A∩B))
• P(B) = P(B\(A∩B)) + P(A∩B)
• P(B) - P(A∩B) = P(B\(A∩B))

условная вероятность и теорема Байеса являются фундаментом для байесовского классификатора

• случайным экспериментом - эксперимент, результат которого не детерминирован изначально
• элементарный исход - исхода случайного эксперимента
• пространством элементарных исходов - омега Ω

P(A) = n/N - вероятность А

• n - равновероятные элементарные исходы, составляющие событие A,
• N - все возможные элементарные исходы

Дополнение события А - это неболагоприятные события для А. У дополнения А и самого А нет общих исходов - они взаимоисключающие, называют несовместные. (черта над А, у нас ^A)

• P(A) - P(^A) = 1
• 1 - 0.7 = 0.3 - 0.7 и 0.3 вероятности 2 взаимоисключающих событий

вероятность двух несовместимых событий в одном эксперименте

• P(A U B ) = P(A) + P(B) - где А и B - несовместимые

вероятность двух независимых событий в двух разных экспериментах или пересечение в одном

• P(A ∩ B) = P(A) * P(B)
• если равенство выполняется, то события независимы

вероятность двух совместимых событий в двух разных экспериментах

• P(A U B) = P(A) + P(B) - P(A ∩ B)

Разбиение вероятностного пространства — это взаимоисключающие и совместно исчерпывающие события

Услованая вероятность P(B|A) - вероятность B при условии A уже произошло (posterior probability)

• P(B|A) = P(A and B) / P(A)
  • где P(A and B) - вероятность что произошло А и затем B
• P(A and B) = P(A) * P(B|A)
  • если это один эксперимент, то скорее всего P(A and B) = P(A)
  • например: Среди студентов-первокурсников онлайн-магистратуры SkillFactory 30 % устроились на работу в первый месяц обучения, а остальные — позже. Среди тех, кто устроился на работу в первый месяц, 70 % получили повышение в течение года.
    • 30/70*70/100=0.3 получили повышение в течении первого года

Precision = P(real=1&pred=1|pred=1) = tp/(tp+fp) Recall = P(real=1&pred=1|real=1) = tp/(tp+fn) Specificity = P(real=0&pred=0|pred=0)

Cобытия A и B называются независимыми, если вероятность их пересечения равна произведению вероятностей

• P(A and B) = P(A) * P(B|A)

Вероятность, что случайно выбранный человек пьёт кофе, равна сумме вероятности, что человек пьёт кофе и является мужчиной, и вероятности, что человек пьет кофе и является женщиной.

Marginal Probability - probability of one event, P(A).

Joint probability - just product ot two Independent events. ex. for two coins: 01 11 10 00

Independent events

• P(A and B) = P(A∩B)=P(A)P(B) - joint probability equals the product of their probabilities
• two coins are flipped the chance of both being heads is 1/2*1/2 = 1/4

Mutually exclusive events

• P(A and B) = P(A∩B)= 0
• P(A or B) = P(A∪B) = P(A) + P(B) - P(A∩B) = P(A)+P(B)
• Ex. Chance of rolling a 1 or 2 on a six-sided die is P(1 or 2) = P(1)+P(2) = 1/6 + 1/6 = 1/3

Not mutually exclusive events

• P(A or B) = P(A∪B) = P(A) + P(B) - P(A∩B)
• Ex. drawing card from a deck of cards, the chance of getting a heart or a face card (J,Q,K) (or one that is both):
  • total 52, 13 - hearts, 12 - face cards, 3 - heart face cards in deck.
  • 13/52 + 12/52 - 3/52 = 11/26

Conditional probability - probability of some event A, given the occurrence [əˈkʌrəns] of some other event B (учитывая наличие)

• P(A|B) - "A given B" - Вероятность A учитывая unconditional B
• Ex. Box: 2 red and 2 blue balls. Taking second boll A pobability: 1/3
• P(B)=1/2 P(A∩B) = x/(1/2) = 1/3
• P(A|B) = P(A∩B)/P(B) = P(B|A)P(A)/P(B)
• P(A and B) = P(A|B)*P(B) - вероятность совместного появления двух зависимых событий А после B:
• Ex. В урне 4 белых и 7 черных шаров. всего 11. Оба белые?
  • P(B) = 4/11
  • P(A|B) = 3/10
  • P(A∩B) = 3/10*4/11=6/55

see 12.3

• P(A|B) = P(A and B)/P(B)
• P(B|A) = P(A and B)/P(A)

Теорема Байеса - позволяет «переставить причину и следствие»

• P(A|B) = ( P(B|A)*P(A) ) / P(B)
• из P(B|A) можно получить P(A|B)
• P(A and B) = P(A) * P(B|A) = P(B) * P(A|B)
• Posterior = ( Likelihood * Prior ) / Evidence

! P(A|B) = P(B|A)*P(A)/P(B) !

• Если B разложить на "B и A" и "B и не A", то:
  • P(B) = P(B,A)+P(B,¬A)
  • P(A|B) = P(B|A)P(A)/(P(B|A)P(A)+ P(B|¬A)P(¬A))

in P(A|B) = ( P(B|A)*P(A) ) / P(B)

• posterior probability of A = P(A|B)
• prior probability of A = P(A)
• A is called hypothesis and B is given data

If P(A|B) = P(A), then events A and B are said to be independent.

Классической или частотная статистика - , которая предполагает, что вероятности — это частота конкретных случайных событий, происходящих в длительном цикле повторяющихся испытаний.

считаем, что случайную величину можно оценить, только если будет проведено большое количество экспериментов.

предоставляет математические инструменты для обновления представлений о случайных событиях в свете появления новых данных или свидетельств об этих событиях.

случайная величина — это детерминированный процесс, который можно просчитать даже без экспериментов, если знать значение всех влияющих на процесс факторов.

• Какова вероятность того, что в результате n независимых испытаний событие A наступит ровно k раз?
• P(A)=p
• Pk =
• Ckn - Количество различных сочетаний успеха и неудач в последовтаельности испытаний
• Вероятность удачной комбинации P(A)P(B) = p^k*(1-p)^n-k
• Pnk=Ckn*p^k*(1-p)^n-k см 11.1.1

• свойства https://en.wikipedia.org/wiki/Probability_distribution
• https://en.wikipedia.org/wiki/List_of_probability_distributions

Распределение - это формула описания значений случайной величины - для каждого события какая вероятность.

чтобы узнать, какое в среднем будет расстояние между двумя случайными точками в круге, методом Монте-Карло, нужно взять много случайных пар точек, для каждой пары найти расстояние, а потом усреднить.

the average of the results obtained from a large number of trials should be close to the expected value and tends to become closer to the expected value as more trials are performed.

• LLN only applies to the average.

It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal distribution as the number of samples increases.

in many situations, for identically distributed independent samples, the standardized sample mean tends towards the standard normal distribution even if the original variables themselves are not normally distributed.

Неформально говоря, классическая центральная предельная теорема утверждает, что сумма n независимых одинаково распределённых случайных величин имеет распределение, близкое к N(n*μ,n*σ^2)

сумма достаточно большого количества слабо зависимых случайных величин, имеющих примерно одинаковые масштабы (ни одно из слагаемых не доминирует, не вносит в сумму определяющего вклада), имеет распределение, близкое к нормальному.

последовательность случайных событий с конечным или счётным числом исходов, где вероятность наступления каждого события зависит только от состояния, достигнутого в предыдущем событии

<2023-04-16 Sun>

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from scipy.interpolate import UnivariateSpline
from math import sqrt
from statistics import variance, stdev

arr = np.random.normal(size=100) # continuous probability mass function
print(pd.Series(arr).describe())

fig = plt.figure(figsize=(8,3))

print(" --------- Probability density function (PDF) or density---------")
print(" - the area under the entire curve is equal to 1")

sp1 = fig.add_subplot(1, 2, 1)   #1 row 2 columns - left
# histogram
sp1.hist(arr, density=True, bins=20)
# line
hist, bin_edges = np.histogram(arr, bins=20, density=True)
x = bin_edges
x = x[:-1] + (x[1] - x[0])/2   # convert bin edges to centers
f = UnivariateSpline(x, hist, k=5)
sp1.plot(x, hist) # line
sp1.plot(x, f(x)) # Spline
sp1.set_title("Probability density function (PDF)")

print(" -------- cumulative distribution function (CDF) ---------")
print(" (x->-∞)lim F(x) = 0 , (x->+∞)lim F(x) = 1")
sp2 = fig.add_subplot(1, 2, 2)   #1 row 2 columns - right

hist, bin_edges = np.histogram(arr, bins=20, density=False)
cumsum = np.cumsum(hist)
cdf = cumsum/cumsum[-1]
sp2.plot(bin_edges[:-1], cdf)
sp2.set_title("cumulative distribution function (CDF)")

# ------ variance -----

print("variance:\t", variance(arr))
# ------ std --------
print("std:\t\t", stdev(arr))

print("sqrt of variance:", sqrt(variance(arr)))

count    100.000000
mean       0.015371
std        1.036308
min       -3.213961
25%       -0.684610
50%       -0.001296
75%        0.618288
max        2.394898
dtype: float64
 --------- Probability density function (PDF) or density---------
 - the area under the entire curve is equal to 1
 -------- cumulative distribution function (CDF) ---------
 (x->-∞)lim F(x) = 0 , (x->+∞)lim F(x) = 1
variance:	 1.073933925916817
std:		 1.0363078335691653
sqrt of variance: 1.0363078335691653

plt.savefig('./autoimgs/pdfcdf.png')

point estimator

is a function of the random sample that is used to estimate an unknown quantity.

• Q = h(X1, X2, X3) , ex. Q = (X1+X2+X3)/n

• Descriptive statistics
• Data collection
• Statistical inference
  • Frequentist inference
  • Bayesian inference
  • Statistical theory - probability distribution, bootstrap, loss function
• Correlation
• Regression analysis
• Categorical / Multivariate / Time-series / Survival analysis

is a collection of methods that deal with drawing conclusions from data that are prone to random variation.

ex: Let X be a normal random variable with mean μ = 100 and variance σ^2 = 15. Find the probability that X>110.

we should conclude whether has a normal distribution or not. Now, suppose that we can use the central limit theorem to argue that is normally distributed.

Approaches to estimate some quantity from the data:

• Frequentist (classical) Inference
  • unknown quantity is assumed to be a fixed quantity. ? = ? /n
  • deals with estimating non-random quantities
  • Maximum Likelihood Estimation (MLE),
• Bayesian Inference
  • deals with estimating random variables.
  • Maximum a Posteriori (MAP),

zero if all the data are the same and increases as the data become more diverse:

• Standard deviation
• Interquartile range (IQR)
• Range
• Mean absolute difference (also known as Gini mean absolute difference)
• Median absolute deviation (MAD)
• Average absolute deviation (or simply called average deviation)
• Distance standard deviation

• statistical population - set of items for experiment.
• sample - subset of population
• sampling fraction = len(sample)/len(population)
• sample points - elements of sample

involves the use of sample date to calculate a signle values - best guess of unknown population paremeter

Rule for calculating an estimate of a given quantity based on observed data

• estimand - θ - fixed parameter of model being estimated
• estimator q - is a function that maps the sample space to a set of sample estimates
• Error - For a given sample - e(x) = q(x)-θ
• MSE of q - E[(q(X)-θ)^2)]

Types:

• point estimator - single value
• interval estimator - two values
  • confidence intervals - Frequentist inference
  • credible intervals - Bayesian inference

Доверительный интервал - покрывает неизвестный параметр с заданной точностью. Чтобы population parameter попадал в интервал в 95 процентах случаев.

более предпочтителен чем точечние оценивание при небольшой выборке

Различают классический (confidence interval CI) и баесовый интервал (credible interval)

Most commonly 95% confidence level used.

Ex: Here is CI interval from XYZ experiments. Means:

confidence interval

If we repeat the experiment infinitely times 95% of calculated intervals will capture parameter in CI. Frequientist. boundaries random, parameter fixed.

credible interval

There is 95% probability/plausibility/likelihood that the parameter lies in the interval. Bayesian statistics. Boundaris fixed, variable random.

confidence interval

If we repeat the experiment infinitely times 95% of calculated intervals will capture parameter in CI. Frequientist.

Понятия вероятности и правдоподобия тесно связаны.

• Вероятность - используется для описания результата функции при наличии фиксированного значения параметра.
• Правдоподобие - используется для описания параметра функции при наличии результата функции.
• «Какова вероятность выпадения 12 очков в каждом из ста бросков двух костей?»
• «Насколько правдоподобно, что кости не шулерские, если из ста бросков в каждом выпало 12 очков?»

Функция правдоподобия Likelihood function

• P(x) = P(x)
• L(q) = L(x=X|q) - показывает, насколько правдоподобно выбранное значение параметра q при известном событии X.

method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis

• statistical inference - статистический вывод process of using data analysis to infer properties of an underlying distribution of probability
• data at hand - имеющиеся данные
• provisional conclusions - предварительные выводы
• statistical independence - probability independence of events
• the null hypothesis (denoted H0) - ('no effect' or 'no difference'). Tested in a test of statistical significance.
• alternative hypothesis (H1 and Ha)
• Test statistic T (Статистический критерий) - математическое правило, в соответствии с которым принимается или отвергается та или иная статистическая гипотеза с заданным уровнем значимости.
  • t-statistic - used in Student's t-test -
  • F-test -

Statistical significance(denoted by α or - when it is very unlikely to have occurred given the null hypothesis (simply by chance alone). The significance level α is the threshold for p below which the null hypothesis is rejected even though by assumption it were true, and something else is going on.

• is the probability of the study rejecting the null hypothesis, given that the null hypothesis is true.
• The significance level for a study is chosen before data collection, and is typically set to 5%
• p-value - the probability of obtaining a result at least as extreme, given that the null hypothesis is true. Как я понял, вычеленная вероятность, которая подтвердила null гипотезу оказавшись меньше The significance level.
• One- and two-tailed tests - способ расчета p-value как хвосты нормального распределения - места где значения маловероятны.

p-value vs α:

• p-value is the probability of obtaining a result at least as extreme, given that the null hypothesis is true.
• a - is the probability of the study rejecting the null hypothesis, given that the null hypothesis is true
• The result is statistically significant, by the standards of the study, when p ≤ α.

Fisher's null hypothesis testing

• Do not use a conventional 5% level
• do not talk about accepting or rejecting hypotheses.
• If the result is "not significant", draw no conclusions and make no decisions, but suspend judgement until further data is available

• To find the number of trials needed to get an effect of a certain size. This is probably the most common use for power analysis–it tells you how many trials you need to do to avoid incorrectly rejecting the null hypothesis.
• To find the power, given an effect size and the number of trials available. This is often useful when you have a limited budget, for say, 100 trials, and you want to know if that number of trials is enough to detect an effect.
• To validate your research. Conducting power analysis is simply put–good science.

creation of new samples based on one observed sample

the proportion of variance in the response variable y that your regression model is able to “explain” via the introduction of regression variables.

• measure in Regression Analysis
• confused with the coefficient of correlation r²
• R² lets you quantify just how much better the Linear model fits the data as compared to the Mean Model.

Total Sum of Square (TSS). ∑(yi-ymean)^2, where ∑ is for i of N.

• quantity that is proportional to the variance in y
• It is the variance that the Mean Model wasn’t able to explain.
• Because TSS/N is the actual variance in y, the TSS is proportional to the total variance in your data.

Mean Model (Null Model) : y_ped = y_mean

Exposure(воздействие) -> Mediator -> Outcome ( Exposure -> Outcome )

• where -> is cause

Exposure <- Confounder -> Outcome ( Exposure -!- Outcome )

There is no confronder if P(y|do(x)) = P(y|x), where P(y|x) is controlled experiment, with x randomized, do(x) - when x is under the hypothetical intervention.

can be obtained by "adjusting" for all confounding factors, namely, conditioning on their various values and averaging the result.

• P(y|do(x)) = ∑P(y|x,z)P(z), where ∑ at z

https://en.wikipedia.org/wiki/Confounding

concerning instantaneous rates of change and slopes of curves - мгновенных скоростей и угловой коэффициент

• Scalar fields - function that associating a single number to every point in a space
• Vector fields - function that associating a vector to each point in a space

Применяются в сжатии звука в MP3, сжатии изображений в JPEG и др.

Сопоставляющая одной функции вещественной переменной другую функцию вещественной переменной.