Math Gamification
Materials for foreign and Russian-speaking students studying the Data Analysis course in English
Probability - Вероятность
The chance of an event occurring, expressed as a number between 0 and 1.
Sample Space - Пространство элементарных событий
The set of all possible outcomes of an experiment.
Event - Событие
A subset of the sample space; a specific outcome or a collection of outcomes.
Random Variable - Случайная величина
A variable whose value is a numerical outcome of a random phenomenon.
Discrete Random Variable - Дискретная случайная величина
A random variable that can only take on a finite number of values or a countably infinite number of values.
Continuous Random Variable - Непрерывная случайная величина
A random variable that can take on any value within a given range.
Permutation - Перестановка
An arrangement of all or part of a set of objects in a specific order. The order of arrangement is crucial. For example, the sequences ABC, ACB, BAC are different permutations. Permutations of all elements nPn = n!
Permutations involving some elements from n: nPx = n!/(n-x)!
Distinct permutation - Уникальная перестановка
Permutation, involving some identical elements: MISSISSIPPI n=11, M=1, I=4, S=4, P=2 n!/(a!b!c!d!)
Combination - Сочетание
A selection of all or part of a set of objects without regard to the order in which they are selected. The order does not matter. For example, the set {A, B, C} is the same as {C, B, A}; they are considered the same combination. nCx = nPx/x! = n!/(x!(n-x)!)
Collectively exhaustive events - Совместно исчерпывающие события
If at least one of these events must occur. That is, their union covers all possible outcomes of the experiment. Definition: For two events AA and BB, they are collectively exhaustive if P(A∪B)=1. Example: When flipping a coin, the events "getting heads" and "getting tails" are collectively exhaustive.
Mutually exclusive events - Взаимоисключающие события
The occurrence of one event prevents the occurrence of the other. Definition: P(A∩B)=0. Example: When flipping a single coin, the events "getting heads" and "getting tails" are mutually exclusive.
Statistically independent - Статистически независимые события
If the occurrence or non-occurrence of one event does not affect the probability of the occurrence or non-occurrence of the other event: P(A*B) = P(A)*P(B) or P(A|B) = P(A) and P(B|A) = P(B). Example: The results of two successive coin flips are independent events.
Probability Distribution - Распределение вероятностей
A function that describes the probability of each possible value of a random variable.
Probability Mass Function (PMF) - Функция распределения вероятностей (для дискретных случайных величин)
A function that gives the probability of a discrete random variable taking on a specific value.
Probability Density Function (PDF) - Функция плотности вероятности (для непрерывных случайных величин)
A function whose integral over a given range gives the probability that a continuous random variable falls within that range.
Cumulative Distribution Function (CDF) - Интегральная функция распределения
A function that gives the probability that a random variable is less than or equal to a given value.
Expected Value (Expectation) - Математическое ожидание
The average value of a random variable over many repetitions of the experiment.
Variance - Дисперсия
A measure of how spread out the values of a random variable are from its expected value.
Standard Deviation - Среднеквадратичное отклонение
The square root of the variance; another measure of the spread of a random variable's values.
Covariance - Ковариация
A measure of the linear relationship between two random variables.
Correlation - Корреляция
A standardized measure of the linear relationship between two random variables, ranging from -1 to +1.
Conditional Probability - Условная вероятность
The probability of an event occurring given that another event has already occurred.
Bayes' Theorem - Теорема Байеса
A theorem that describes how to update the probability of an event based on new evidence.
Independence - Независимость
Two events are independent if the occurrence of one does not affect the probability of the other.
Mutually Exclusive Events - Взаимоисключающие события
Two events that cannot both occur at the same time.
Random Sample - Случайная выборка
A subset of a population selected in such a way that each member of the population has an equal chance of being selected.
Central Limit Theorem - Центральная предельная теорема
A theorem stating that the distribution of the sample mean approaches a normal distribution as the sample size increases.
Normal Distribution - Нормальное распределение
A bell-shaped probability distribution that is symmetric around its mean.
Standard Normal Distribution - Стандартное нормальное распределение
A normal distribution with a mean of 0 and a standard deviation of 1.
Descriptive Statistics - Описательная статистика
Methods for summarizing and presenting data, such as measures of central tendency and dispersion.
Null Hypothesis H0 - Нулевая гипотеза
The statement being tested in hypothesis testing; often a statement of no effect or no difference.
Alternative Hypothesis H1 - Альтернативная гипотеза
The statement that is accepted if the null hypothesis is rejected.
Type I Error - Ошибка первого рода
Rejecting the null hypothesis when it is true.
Type II Error - Ошибка второго рода
Failing to reject the null hypothesis when it is false.
Hypothesis Testing - Проверка гипотез
A statistical method used to make inferences about a population based on sample data.
Hypothesis Testing (One-tailed) - Проверка гипотезы (односторонняя)
A hypothesis test where the alternative hypothesis specifies a direction (e.g., greater than or less than).
Hypothesis Testing (Two-tailed) - Проверка гипотезы (двусторонняя)
A hypothesis test where the alternative hypothesis does not specify a direction (e.g., not equal to).
Outlier - Выброс
A data point that significantly deviates from other data points in a dataset. Can influence statistical analyses and should be carefully considered.
Confidence Interval - Доверительный интервал
A range of values that is likely to contain the true value of a population parameter.
Significance Level - Уровень значимости
The probability of rejecting a true null hypothesis.
p-value - р-значение
The probability of observing the data (or more extreme data) if the null hypothesis is true.
Student's t-distribution - Распределение Стьюдента
A probability distribution used to estimate population parameters when the sample size is small or the population standard deviation is unknown.
Chi-squared distribution χ² - Распределение хи-квадрат
A probability distribution used in hypothesis testing to determine if there is a significant difference between observed and expected frequencies. Often used in goodness-of-fit tests and tests of independence.
Degrees of Freedom - Число степеней свободы
The number of independent pieces of information available to estimate a parameter. Influences the shape of t-distributions and chi-squared distributions.
F-distribution - F-распределение
A probability distribution used to compare the variances of two or more populations or to test the overall significance of a regression model.
Elective course \ Курс по выбору
Game Theory - Теория игр
The study of mathematical models of strategic interaction among rational agents.
Game - Игра
A situation of strategic interaction between two or more players.
Players - Игроки
The decision-makers in a game.
Strategies - Стратегии
The complete plan of action for a player in a game, specifying actions for all possible contingencies.
Payoffs - Выплаты
The outcomes or consequences of a game for each player, often represented numerically.
Normal Form Game - Игра в нормальной форме
A representation of a game where players' strategies and payoffs are displayed in a matrix or table.
Extensive Form Game - Игра в развернутой форме
A representation of a game that shows the sequence of moves, information sets, and payoffs. Often depicted as a game tree.
Nash Equilibrium - Равновесие Нэша
A set of strategies, one for each player, such that no player has an incentive to deviate unilaterally, given the strategies of the other players.
Pure Strategy - Чистая стратегия
A strategy where a player chooses a single action with certainty.
Mixed Strategy - Смешанная стратегия
A strategy where a player chooses actions probabilistically.
Dominant Strategy - Доминирующая стратегия
A strategy that is always better for a player, regardless of the actions of the other players.
Dominated Strategy - Доминируемая стратегия
A strategy that is always worse for a player than another strategy, regardless of the actions of the other players.
Prisoner's Dilemma - Дилемма заключенных
A classic game theory example illustrating the conflict between individual rationality and collective rationality.
Zero-Sum Game - Игра с нулевой суммой
A game where the total payoffs to all players always sum to zero. One player's gain is always another player's loss.
Non-Zero-Sum Game - Игра с ненулевой суммой
A game where the sum of payoffs to all players is not always zero. Cooperation can lead to mutually beneficial outcomes
Cooperative Game - Кооперативная игра
A game where players can form binding agreements.
Non-Cooperative Game - Некооперативная игра
A game where players cannot form binding agreements.
Game Tree - Дерево игры
A graphical representation of an extensive-form game.
Information Set - Информационное множество
A set of nodes in a game tree that a player cannot distinguish between.
Perfect Information - Совершенная информация
A game where all players know the history of the game at every decision point.
Imperfect Information - Несовершенная информация
A game where some players lack complete information about the history of the game at some decision points.
Subgame Perfect Equilibrium - Совершенное равновесие по подыграм
A Nash equilibrium where every subgame is also a Nash equilibrium.
Repeated Game - Повторяющаяся игра
A game played multiple times.
Backward Induction - Обратная индукция
A method of solving extensive-form games by working backward from the end of the game.
Bayesian Game - Байесовская игра
A game where players have incomplete information about the other players' types or payoffs.
Signaling Game - Игра с сигнализацией
A game where one player sends a signal to the other player, who then takes an action based on the signal.
Auction - Аукцион
A game where bidders compete to purchase an item.
