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  2. Line integral - Wikipedia

    en.wikipedia.org/wiki/Line_integral

    A line integral of a scalar field is thus a line integral of a vector field, where the vectors are always tangential to the line of the integration. Line integrals of vector fields are independent of the parametrization r in absolute value, but they do depend on its orientation. Specifically, a reversal in the orientation of the parametrization ...

  3. Gaussian integral - Wikipedia

    en.wikipedia.org/wiki/Gaussian_integral

    A different technique, which goes back to Laplace (1812), [3] is the following. Let = =. Since the limits on s as y → ±∞ depend on the sign of x, it simplifies the calculation to use the fact that e −x 2 is an even function, and, therefore, the integral over all real numbers is just twice the integral from zero to infinity.

  4. Normal distribution - Wikipedia

    en.wikipedia.org/wiki/Normal_distribution

    The simplest case of a normal distribution is known as the standard normal distribution or unit normal distribution. This is a special case when μ = 0 {\textstyle \mu =0} and σ 2 = 1 {\textstyle \sigma ^{2}=1} , and it is described by this probability density function (or density): φ ( z ) = e − z 2 2 2 π . {\displaystyle \varphi (z ...

  5. Shape of a probability distribution - Wikipedia

    en.wikipedia.org/wiki/Shape_of_a_probability...

    The shape of a distribution will fall somewhere in a continuum where a flat distribution might be considered central and where types of departure from this include: mounded (or unimodal), U-shaped, J-shaped, reverse-J shaped and multi-modal. [1] A bimodal distribution would have two high points rather than one. The shape of a distribution is ...

  6. Bell-shaped function - Wikipedia

    en.wikipedia.org/wiki/Bell-shaped_function

    These functions are typically continuous or smooth, asymptotically approach zero for large negative/positive x, and have a single, unimodal maximum at small x. Hence, the integral of a bell-shaped function is typically a sigmoid function. Bell shaped functions are also commonly symmetric. Many common probability distribution functions are bell ...

  7. Multivariate normal distribution - Wikipedia

    en.wikipedia.org/wiki/Multivariate_normal...

    Bivariate normal distribution centered at (,) with a standard deviation of 3 in roughly the (,) direction and of 1 in the orthogonal direction. As the absolute value of the correlation parameter increases, these loci are squeezed toward the following line :

  8. Stable distribution - Wikipedia

    en.wikipedia.org/wiki/Stable_distribution

    The normal distribution defines a family of stable distributions. By the classical central limit theorem the properly normed sum of a set of random variables, each with finite variance, will tend toward a normal distribution as the number of variables increases. Without the finite variance assumption, the limit may be a stable distribution that ...

  9. Gradient theorem - Wikipedia

    en.wikipedia.org/wiki/Gradient_theorem

    The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve.