|
Continuity Correction
In mathematics, a continuity correction is an adjustment made when a discrete object is approximated using a continuous object. Examples Binomial If a random variable ''X'' has a binomial distribution with parameters ''n'' and ''p'', i.e., ''X'' is distributed as the number of "successes" in ''n'' independent Bernoulli trials with probability ''p'' of success on each trial, then :P(X\leq x) = P(X |
|
 |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
 |
Poisson Distribution
In probability theory and statistics, the Poisson distribution () is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate and independently of the time since the last event. It can also be used for the number of events in other types of intervals than time, and in dimension greater than 1 (e.g., number of events in a given area or volume). The Poisson distribution is named after French mathematician Siméon Denis Poisson. It plays an important role for discrete-stable distributions. Under a Poisson distribution with the expectation of ''λ'' events in a given interval, the probability of ''k'' events in the same interval is: :\frac . For instance, consider a call center which receives an average of ''λ ='' 3 calls per minute at all times of day. If the calls are independent, receiving one does not change the probability of when the next on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
 |
Binomial Proportion Confidence Interval
In statistics, a binomial proportion confidence interval is a confidence interval for the probability of success calculated from the outcome of a series of success–failure experiments (Bernoulli trials). In other words, a binomial proportion confidence interval is an interval estimate of a success probability \ p\ when only the number of experiments \ n\ and the number of successes \ n_\mathsf\ are known. There are several formulas for a binomial confidence interval, but all of them rely on the assumption of a binomial distribution. In general, a binomial distribution applies when an experiment is repeated a fixed number of times, each trial of the experiment has two possible outcomes (success and failure), the probability of success is the same for each trial, and the trials are statistically independent. Because the binomial distribution is a discrete probability distribution (i.e., not continuous) and difficult to calculate for large numbers of trials, a variety of approxima ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
Yates's Correction For Continuity
In statistics, Yates's correction for continuity (or Yates's chi-squared test) is a statistical test commonly used when analyzing count data organized in a contingency table, particularly when sample sizes are small. It is specifically designed for testing whether two categorical variables are related or independent of each other. The correction modifies the standard chi-squared test to account for the fact that a continuous distribution ( chi-squared) is used to approximate discrete data. Almost exclusively applied to 2×2 contingency tables, it involves subtracting 0.5 from the absolute difference between observed and expected frequencies before squaring the result. Unlike the standard Pearson chi-squared statistic, Yates's correction is approximately unbiased for small sample sizes. It is considered more conservative than the uncorrected chi-squared test, as it increases the p-value and thus reduces the likelihood of rejecting the null hypothesis when it is true. While widely ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
 |
Checking Whether A Coin Is Fair
In statistics, the question of checking whether a coin is fair is one whose importance lies, firstly, in providing a simple problem on which to illustrate basic ideas of statistical inference and, secondly, in providing a simple problem that can be used to compare various competing methods of statistical inference, including decision theory. The practical problem of checking whether a coin is fair might be considered as easily solved by performing a sufficiently large number of trials, but statistics and probability theory can provide guidance on two types of question; specifically those of how many trials to undertake and of the accuracy of an estimate of the probability of turning up heads, derived from a given sample of trials. A fair coin is an idealized randomizing device with two states (usually named "heads" and "tails") which are equally likely to occur. It is based on the coin flip used widely in sports and other situations where it is required to give two parties the sa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
Binomial Test
Binomial test is an exact test of the statistical significance of deviations from a theoretically expected distribution of observations into two categories using sample data. Usage A binomial test is a statistical hypothesis test used to determine whether the proportion of successes in a sample differs from an expected proportion in a binomial distribution. It is useful for situations when there are two possible outcomes (e.g., success/failure, yes/no, heads/tails), i.e., where repeated experiments produce binary data. If one assumes an underlying probability \pi_0 between 0 and 1, the null hypothesis is : H_0\colon\pi=\pi_0 For a sample of size n, we would expect n\pi_0 successes. The formula of the binomial distribution gives the probability of those n samples instead producing k successes: : \Pr(X=k)=\binom\pi_0^k(1-\pi_0)^ Suppose that we want to test the alternative hypothesis : H_\colon\pi\pi_0 using the summation of the range from k to n instead. Calculating a p-val ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
 |
Statistical Hypothesis Test
A statistical hypothesis test is a method of statistical inference used to decide whether the data provide sufficient evidence to reject a particular hypothesis. A statistical hypothesis test typically involves a calculation of a test statistic. Then a decision is made, either by comparing the test statistic to a critical value or equivalently by evaluating a ''p''-value computed from the test statistic. Roughly 100 specialized statistical tests are in use and noteworthy. History While hypothesis testing was popularized early in the 20th century, early forms were used in the 1700s. The first use is credited to John Arbuthnot (1710), followed by Pierre-Simon Laplace (1770s), in analyzing the human sex ratio at birth; see . Choice of null hypothesis Paul Meehl has argued that the epistemological importance of the choice of null hypothesis has gone largely unacknowledged. When the null hypothesis is predicted by theory, a more precise experiment will be a more severe test of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
Statistical Software
The following is a list of statistical software. Open-source * ADaMSoft – a generalized statistical software with data mining algorithms and methods for data management * ADMB – a software suite for non-linear statistical modeling based on C++ which uses automatic differentiation * Chronux – for neurobiological time series data * DAP (software), DAP – free replacement for SAS * Environment for DeveLoping KDD-Applications Supported by Index-Structures (ELKI) a software framework for developing data mining algorithms in Java (programming language), Java * Epi Info – List of statistical packages, statistical software for epidemiology developed by Centers for Disease Control and Prevention (CDC). Apache 2 licensed * Fityk – nonlinear regression software (GUI and command line) * GNU Octave – programming language very similar to MATLAB with statistical features * gretl – gnu regression, econometrics and time-series library * intrinsic Noise Analyzer (iNA) – For analyzin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
 |
Variance
In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. 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 \sigma^2, s^2, \operatorname(X), V(X), or \mathbb(X). An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviation; for example, the variance of a sum of uncorrelated random variables is equal to the sum of their variances. A disadvantage of the variance for practical applications is that, unlike the standard deviation, its units differ from the random variable, which is why the standard devi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
 |
Discrete Mathematics
Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous functions). Objects studied in discrete mathematics include integers, Graph (discrete mathematics), graphs, and Statement (logic), statements in Mathematical logic, logic. By contrast, discrete mathematics excludes topics in "continuous mathematics" such as real numbers, calculus or Euclidean geometry. Discrete objects can often be enumeration, enumerated by integers; more formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets (finite sets or sets with the same cardinality as the natural numbers). However, there is no exact definition of the term "discrete mathematics". The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
 |
Expected Value
In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean, mean of the possible values a random variable can take, weighted by the probability of those outcomes. Since it is obtained through arithmetic, the expected value sometimes may not even be included in the sample data set; it is not the value you would expect to get in reality. The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes. In the case of a continuum of possible outcomes, the expectation is defined by Integral, integration. In the axiomatic foundation for probability provided by measure theory, the expectation is given by Lebesgue integration. The expected value of a random variable is often denoted by , , or , with a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
 |
Normal Distribution
In probability theory and 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 f(x) = \frac e^\,. The parameter is the mean or expectation of the distribution (and also its median and mode), while the parameter \sigma^2 is the variance. The standard deviation of the distribution is (sigma). A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. 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 c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |