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In probability theory , a normal or Gaussian or Gauss or Laplace—Gauss distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is. 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.
Skip to search form Skip to main content You are currently offline. Some features of the site may not work correctly. We introduce the theory of distributions and examine their relation to the Fourier transform. We then use this machinery to find solutions to linear partial differential equations, in particular, fundamental solutions to partial differential operators. Finally, we develop Sobolev spaces in order to study the relationship between the regularity of a partial differential equation and its solution, namely elliptic regularity. Save to Library.
Skip to main content Skip to table of contents. Advertisement Hide. This service is more advanced with JavaScript available. Pages Test Functions. Definition and Basic Properties of Distributions.
In probability theory and statistics , the characteristic function of any real-valued random variable completely defines its probability distribution. If a random variable admits a probability density function , then the characteristic function is the Fourier transform of the probability density function. Thus it provides an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution functions. There are particularly simple results for the characteristic functions of distributions defined by the weighted sums of random variables. In addition to univariate distributions , characteristic functions can be defined for vector- or matrix-valued random variables, and can also be extended to more generic cases. The characteristic function always exists when treated as a function of a real-valued argument, unlike the moment-generating function. There are relations between the behavior of the characteristic function of a distribution and properties of the distribution, such as the existence of moments and the existence of a density function.
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On this page :. The aim of the course is to give an introduction to distribution theory, which is an important tool for the theory of partial differential equations. The course treats the foundations of distribution theory: test functions, the concept of a distribution, distributions with compact support, operations on distributions, convolution, homogeneous distributions and the Fourier transform.
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