- published: 12 Dec 2012
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In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the form:
for arbitrary real constants a, b and c. It is named after the mathematician Carl Friedrich Gauss.
The graph of a Gaussian is a characteristic symmetric "bell curve" shape. The parameter a is the height of the curve's peak, b is the position of the center of the peak and c (the standard deviation, sometimes called the Gaussian RMS width) controls the width of the "bell".
Gaussian functions are widely used in statistics where they describe the normal distributions, in signal processing where they serve to define Gaussian filters, in image processing where two-dimensional Gaussians are used for Gaussian blurs, and in mathematics where they are used to solve heat equations and diffusion equations and to define the Weierstrass transform.
Gaussian functions arise by composing the exponential function with a concave quadratic function. The Gaussian functions are thus those functions whose logarithm is a concave quadratic function.
Khan Academy is a non-profit educational organization created in 2006 by educator Salman Khan with the aim of providing a free, world-class education for anyone, anywhere. The organization produces short lectures in the form of YouTube videos. In addition to micro lectures, the organization's website features practice exercises and tools for educators. All resources are available for free to anyone around the world. The main language of the website is English, but the content is also available in other languages.
The founder of the organization, Salman Khan, was born in New Orleans, Louisiana, United States to immigrant parents from Bangladesh and India. After earning three degrees from the Massachusetts Institute of Technology (a BS in mathematics, a BS in electrical engineering and computer science, and an MEng in electrical engineering and computer science), he pursued an MBA from Harvard Business School.
In late 2004, Khan began tutoring his cousin Nadia who needed help with math using Yahoo!'s Doodle notepad.When other relatives and friends sought similar help, he decided that it would be more practical to distribute the tutorials on YouTube. The videos' popularity and the testimonials of appreciative students prompted Khan to quit his job in finance as a hedge fund analyst at Connective Capital Management in 2009, and focus on the tutorials (then released under the moniker "Khan Academy") full-time.
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The most widely used probability distribution function is the Gaussian curve, also known as the normal distribution or the bell curve. In this video I introduce the Gaussian, and its integral.
In probability theory, the normal (or Gaussian) distribution is a very commonly occurring continuous probability distribution—a function that tells the probability of a number in some context falling between any two real numbers. For example, the distribution of income measured on a log scale is normally distributed in some contexts, as is often the distribution of grades on a test administered to many people. Normal distributions are extremely important in statistics and are often used in the natural and social sciences for real-valued random variables whose distributions are not known. The normal distribution is immensely useful because of the central limit theorem, which states that, under mild conditions, the mean of many random variables independently drawn from the same distribution ...
A clever way to evaluate the Gaussian Integral analytically. A great math trick to show your friends.
In mathematics, a Gaussian function (named after Carl Friedrich Gauss) is a function of the form: This video is targeted to blind users. Attribution: Article text available under CC-BY-SA Creative Commons image source in video
Fourier transform of Gaussian function is discussed in this lecture. Fourier transform of Gaussian function is another Gaussian function.
This video lecture explains how one can solve the integration using the properties of Gaussian probability density function.
MSE101 Mathematics - Data Analysis Lecture 4.1 - Integrating the Gaussian between limits - the erf function Course webpage with notes: http://dyedavid.com/mse101 Dr Sam Cooper, Department of Materials, Imperial College, London, UK
The gaussian blur algorithm is one of the most widely used blurring algorithms. It is accomplished by applying a convolution kernel to every pixel of an image, and averaging each value of each color channel of each pixel with the corresponding elements of the convolution matrix. You can also weigh the kernel so that each pixel processed takes a fraction of its neighboring pixels instead of the whole value. LIVE DEMO: http://easylearntutorial.com/live-demo/gaussian-blur-image-processing-algorithm.php The algorithm (source: WikiPedia) The Big-O value for the Gaussian blur algorithm is O(Kw * Kh * Iw * Ih), where K[w,h] and I[w,h] are the width and height of the kernel and image, respectively. Programming tutorials by Easy Learn Tutorial - because anyone can learn how to become an expert...
Probability density functions for continuous random variables. Practice this yourself on Khan Academy right now: https://www.khanacademy.org/e/probability-models?utm_source=YTdescription&utm;_medium=YTdescription&utm;_campaign=YTdescription Watch the next lesson: https://www.khanacademy.org/math/probability/random-variables-topic/expected-value/v/term-life-insurance-and-death-probability?utm_source=YT&utm;_medium=Desc&utm;_campaign=ProbabilityandStatistics Missed the previous lesson? https://www.khanacademy.org/math/probability/random-variables-topic/random_variables_prob_dist/v/discrete-probability-distribution?utm_source=YT&utm;_medium=Desc&utm;_campaign=ProbabilityandStatistics Probability and statistics on Khan Academy: We dare you to go through a day in which you never consider or use ...
The Gaussian function has an important role in PDEs and so we go over direct computation of the this function