Logistic regressionIn statistics, the logistic model (or logit model) is a statistical model that models the probability of an event taking place by having the log-odds for the event be a linear combination of one or more independent variables. In regression analysis, logistic regression (or logit regression) is estimating the parameters of a logistic model (the coefficients in the linear combination).
Normal distributionIn statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is The parameter is the mean or expectation of the distribution (and also its median and mode), while the parameter is its standard deviation. The variance of the distribution is . A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate.
Cauchy distributionThe Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) function, or Breit–Wigner distribution. The Cauchy distribution is the distribution of the x-intercept of a ray issuing from with a uniformly distributed angle. It is also the distribution of the ratio of two independent normally distributed random variables with mean zero.
Discrete choiceIn economics, discrete choice models, or qualitative choice models, describe, explain, and predict choices between two or more discrete alternatives, such as entering or not entering the labor market, or choosing between modes of transport. Such choices contrast with standard consumption models in which the quantity of each good consumed is assumed to be a continuous variable. In the continuous case, calculus methods (e.g. first-order conditions) can be used to determine the optimum amount chosen, and demand can be modeled empirically using regression analysis.
Chi-squared distributionIn probability theory and statistics, the chi-squared distribution (also chi-square or -distribution) with degrees of freedom is the distribution of a sum of the squares of independent standard normal random variables. The chi-squared distribution is a special case of the gamma distribution and is one of the most widely used probability distributions in inferential statistics, notably in hypothesis testing and in construction of confidence intervals.
Stable distributionIn probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be stable if its distribution is stable. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it. Of the four parameters defining the family, most attention has been focused on the stability parameter, (see panel).
Ratio distributionA ratio distribution (also known as a quotient distribution) is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions. Given two (usually independent) random variables X and Y, the distribution of the random variable Z that is formed as the ratio Z = X/Y is a ratio distribution. An example is the Cauchy distribution (also called the normal ratio distribution), which comes about as the ratio of two normally distributed variables with zero mean.
F-distributionIn probability theory and statistics, the F-distribution or F-ratio, also known as Snedecor's F distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor), is a continuous probability distribution that arises frequently as the null distribution of a test statistic, most notably in the analysis of variance (ANOVA) and other F-tests. The F-distribution with d1 and d2 degrees of freedom is the distribution of where and are independent random variables with chi-square distributions with respective degrees of freedom and .
Gumbel distributionIn probability theory and statistics, the Gumbel distribution (also known as the type-I generalized extreme value distribution) is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. This distribution might be used to represent the distribution of the maximum level of a river in a particular year if there was a list of maximum values for the past ten years. It is useful in predicting the chance that an extreme earthquake, flood or other natural disaster will occur.
Random variableA random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' can be misleading as it is not actually random nor a variable, but rather it is a function from possible outcomes (e.g., the possible upper sides of a flipped coin such as heads and tails ) in a sample space (e.g., the set ) to a measurable space (e.g., in which 1 corresponding to and −1 corresponding to ), often to the real numbers.
Logistic distributionIn probability theory and statistics, the logistic distribution is a continuous probability distribution. Its cumulative distribution function is the logistic function, which appears in logistic regression and feedforward neural networks. It resembles the normal distribution in shape but has heavier tails (higher kurtosis). The logistic distribution is a special case of the Tukey lambda distribution.
Binary regressionIn statistics, specifically regression analysis, a binary regression estimates a relationship between one or more explanatory variables and a single output binary variable. Generally the probability of the two alternatives is modeled, instead of simply outputting a single value, as in linear regression. Binary regression is usually analyzed as a special case of binomial regression, with a single outcome (), and one of the two alternatives considered as "success" and coded as 1: the value is the count of successes in 1 trial, either 0 or 1.
VarianceIn probability theory and statistics, variance is the squared deviation from the mean of a random variable. The variance is also often defined as the square of the standard deviation. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. It is the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by , , , , or .
Statistical assumptionStatistics, like all mathematical disciplines, does not infer valid conclusions from nothing. Inferring interesting conclusions about real statistical populations almost always requires some background assumptions. Those assumptions must be made carefully, because incorrect assumptions can generate wildly inaccurate conclusions. Here are some examples of statistical assumptions: Independence of observations from each other (this assumption is an especially common error). Independence of observational error from potential confounding effects.
Generalized extreme value distributionIn probability theory and statistics, the generalized extreme value (GEV) distribution is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known as type I, II and III extreme value distributions. By the extreme value theorem the GEV distribution is the only possible limit distribution of properly normalized maxima of a sequence of independent and identically distributed random variables.
Complex random variableIn probability theory and statistics, complex random variables are a generalization of real-valued random variables to complex numbers, i.e. the possible values a complex random variable may take are complex numbers. Complex random variables can always be considered as pairs of real random variables: their real and imaginary parts. Therefore, the distribution of one complex random variable may be interpreted as the joint distribution of two real random variables.
Exchangeable random variablesIn statistics, an exchangeable sequence of random variables (also sometimes interchangeable) is a sequence X1, X2, X3, ... (which may be finitely or infinitely long) whose joint probability distribution does not change when the positions in the sequence in which finitely many of them appear are altered. Thus, for example the sequences both have the same joint probability distribution. It is closely related to the use of independent and identically distributed random variables in statistical models.
LogitIn statistics, the logit (ˈloʊdʒɪt ) function is the quantile function associated with the standard logistic distribution. It has many uses in data analysis and machine learning, especially in data transformations. Mathematically, the logit is the inverse of the standard logistic function , so the logit is defined as Because of this, the logit is also called the log-odds since it is equal to the logarithm of the odds where p is a probability. Thus, the logit is a type of function that maps probability values from to real numbers in , akin to the probit function.
Convergence of random variablesIn probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes. The same concepts are known in more general mathematics as stochastic convergence and they formalize the idea that a sequence of essentially random or unpredictable events can sometimes be expected to settle down into a behavior that is essentially unchanging when items far enough into the sequence are studied.
Independent and identically distributed random variablesIn probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usually abbreviated as i.i.d., iid, or IID. IID was first defined in statistics and finds application in different fields such as data mining and signal processing. Statistics commonly deals with random samples. A random sample can be thought of as a set of objects that are chosen randomly.
