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 picture info Expected Value In probability theory, the expected value of a random variable, intuitively, is the long-run average value of repetitions of the experiment it represents. For example, the expected value in rolling a six-sided dice is 3.5, because the average of all the numbers that come up in an extremely large number of rolls is close to 3.5. Less roughly, the law of large numbers states that the arithmetic mean of the values almost surely converges to the expected value as the number of repetitions approaches infinity. The expected value is also known as the expectation, mathematical expectation, EV, average, mean value, mean, or first moment. More practically, the expected value of a discrete random variable is the probability-weighted average of all possible values. In other words, each possible value the random variable can assume is multiplied by its probability of occurring, and the resulting products are summed to produce the expected value [...More Info...]       [...Related Items...] Monotone Convergence Theorem In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the convergence of monotonic sequences (sequences that are increasing or decreasing) that are also bounded [...More Info...]       [...Related Items...] picture info Convergent Sequence As the positive integer ${\displaystyle n}$ becomes larger and larger, the value ${\displaystyle n\cdot \sin {\bigg (}{\frac {1}{n}}{\bigg )}}$ becomes arbitrarily close to ${\displaystyle 1}$ [...More Info...]       [...Related Items...] picture info Security Breach Security is freedom from, or resilience against, potential harm (or other unwanted coercive change) from external forces. Beneficiaries (technically referents) of security may be persons and social groups, objects and institutions, ecosystems, and any other entity or phenomenon vulnerable to unwanted change by its environment. Refugees fleeing war and insecurity in Iraq and Syria arrive at Lesbos Island, supported by Spanish volunteers, 2015 Security mostly refers to protection from hostile forces, but it has a wide range of other senses: for example, as the absence of harm (e.g. freedom from want); as the presence of an essential good (e.g. food security); as resilience against potential damage or harm (e.g. secure foundations); as secrecy (e.g. a secure telephone line); as containment (e.g. a secure room or cell); and as a state of mind (e.g [...More Info...]       [...Related Items...] picture info Riemann Rearrangement Theorem Georg Friedrich Bernhard Riemann (German: [ˈʀiːman] (); 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis. His famous 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, is regarded as one of the most influential papers in analytic number theory [...More Info...]       [...Related Items...] picture info Dependent Variable In mathematical modeling, statistical modeling and experimental sciences, the values of dependent variables depend on the values of independent variables. The dependent variables represent the output or outcome whose variation is being studied. The independent variables represent inputs or causes, i.e., potential reasons for variation or, in the experimental setting, the variable controlled by the experimenter. Models and experiments test or determine the effects that the independent variables have on the dependent variables [...More Info...]       [...Related Items...] picture info Explanatory Variable In mathematical modeling, statistical modeling and experimental sciences, the values of dependent variables depend on the values of independent variables. The dependent variables represent the output or outcome whose variation is being studied. The independent variables represent inputs or causes, i.e., potential reasons for variation or, in the experimental setting, the variable controlled by the experimenter. Models and experiments test or determine the effects that the independent variables have on the dependent variables [...More Info...]       [...Related Items...] picture info Regression Analysis In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships among variables. It includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables (or 'predictors'). More specifically, regression analysis helps one understand how the typical value of the dependent variable (or 'criterion variable') changes when any one of the independent variables is varied, while the other independent variables are held fixed. Most commonly, regression analysis estimates the conditional expectation of the dependent variable given the independent variables – that is, the average value of the dependent variable when the independent variables are fixed. Less commonly, the focus is on a quantile, or other location parameter of the conditional distribution of the dependent variable given the independent variables [...More Info...]       [...Related Items...] picture info Statistical Dispersion In statistics, dispersion (also called variability, scatter, or spread) is the extent to which a distribution is stretched or squeezed. Common examples of measures of statistical dispersion are the variance, standard deviation, and interquartile range. Dispersion is contrasted with location or central tendency, and together they are the most used properties of distributions. [...More Info...]       [...Related Items...] picture info Infinite Sum In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics), through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For a long time, the idea that such a potentially infinite summation could produce a finite result was considered paradoxical by mathematicians and philosophers. This paradox was resolved using the concept of a limit during the 19th century [...More Info...]       [...Related Items...] picture info Improper Integral In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number, ${\displaystyle \infty }$, ${\displaystyle -\infty }$, or in some instances as both endpoints approach limits [...More Info...]       [...Related Items...]