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List Of Integrals Of Gaussian Functions
In the expressions in this article, :\phi(x) = \frace^ is the standard normal probability density function, :\Phi(x) = \int_^x \phi(t) \, dt = \frac\left(1 + \operatorname\left(\frac\right)\right) is the corresponding cumulative distribution function (where erf is the error function) and : T(h,a) = \phi(h)\int_0^a \frac \, dx is Owen's T function. Owen has an extensive list of Gaussian-type integrals; only a subset is given below. Indefinite integrals :\int \phi(x) \, dx = \Phi(x) + C :\int x \phi(x) \, dx = -\phi(x) + C :\int x^2 \phi(x) \, dx = \Phi(x) - x\phi(x) + C :\int x^ \phi(x) \, dx = -\phi(x) \sum_^k \fracx^ + C lists this integral above without the minus sign, which is an error. See calculation bWolframAlpha/ref> :\int x^ \phi(x) \, dx = -\phi(x)\sum_^k\fracx^ + (2k+1)!!\,\Phi(x) + C In these integrals, ''n''!! is the double factorial In mathematics, the double factorial or semifactorial of a number , denoted by , is the product of all the integers from 1 u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Standard Normal
In 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 \mu is the mean or expectation of the distribution (and also its median and mode), while the parameter \sigma is its standard deviation. The variance of the distribution is \sigma^2. 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 converges to a normal distrib ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cumulative Distribution Function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Every probability distribution supported on the real numbers, discrete or "mixed" as well as continuous, is uniquely identified by an ''upwards continuous'' ''monotonic increasing'' cumulative distribution function F : \mathbb R \rightarrow ,1/math> satisfying \lim_F(x)=0 and \lim_F(x)=1. In the case of a scalar continuous distribution, it gives the area under the probability density function from minus infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables. Definition The cumulative distribution function of a real-valued random variable X is the function given by where the right-hand side represents the probability that the random variable X takes on a value less ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Error Function
In mathematics, the error function (also called the Gauss error function), often denoted by , is a complex function of a complex variable defined as: :\operatorname z = \frac\int_0^z e^\,\mathrm dt. This integral is a special (non- elementary) sigmoid function that occurs often in probability, statistics, and partial differential equations. In many of these applications, the function argument is a real number. If the function argument is real, then the function value is also real. In statistics, for non-negative values of , the error function has the following interpretation: for a random variable that is normally distributed with mean 0 and standard deviation , is the probability that falls in the range . Two closely related functions are the complementary error function () defined as :\operatorname z = 1 - \operatorname z, and the imaginary error function () defined as :\operatorname z = -i\operatorname iz, where is the imaginary unit Name The name "error functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Owen's T Function
In mathematics, Owen's T function ''T''(''h'', ''a''), named after statistician Donald Bruce Owen, is defined by : T(h,a)=\frac\int_^ \frac dx \quad \left(-\infty < h, a < +\infty\right). The function was first introduced by Owen in 1956. Applications The function ''T''(''h'', ''a'') gives the probability of the event (''X'' > ''h'' and 0 < ''Y'' < ''aX'') where ''X'' and ''Y'' are s. This function can be used to calculate |
Double Factorial
In mathematics, the double factorial or semifactorial of a number , denoted by , is the product of all the integers from 1 up to that have the same parity (odd or even) as . That is, :n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots. For even , the double factorial is :n!! = \prod_^\frac (2k) = n(n-2)(n-4)\cdots 4\cdot 2 \,, and for odd it is :n!! = \prod_^\frac (2k-1) = n(n-2)(n-4)\cdots 3\cdot 1 \,. For example, . The zero double factorial as an empty product. The sequence of double factorials for even = starts as : 1, 2, 8, 48, 384, 3840, 46080, 645120,... The sequence of double factorials for odd = starts as : 1, 3, 15, 105, 945, 10395, 135135,... The term odd factorial is sometimes used for the double factorial of an odd number. History and usage In a 1902 paper, the physicist Arthur Schuster wrote: states that the double factorial was originally introduced in order to simplify the expression of certain trigonometric integrals that arise in the derivation of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lists Of Integrals
Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. This page lists some of the most common antiderivatives. Historical development of integrals A compilation of a list of integrals (Integraltafeln) and techniques of integral calculus was published by the German mathematician (aka ) in 1810. These tables were republished in the United Kingdom in 1823. More extensive tables were compiled in 1858 by the Dutch mathematician David Bierens de Haan for his ''Tables d'intégrales définies'', supplemented by ''Supplément aux tables d'intégrales définies'' in ca. 1864. A new edition was published in 1867 under the title ''Nouvelles tables d'intégrales définies''. These tables, which contain mainly integrals of elementary functions, remained in u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
