Gaussian Normal Error Curve
Contents |
be challenged and removed. (August 2009) (Learn how and when to remove this template message) Normalized Gaussian curves with expected value μ and variance σ2. gaussian distribution function The corresponding parameters are a = 1 σ 2 π {\displaystyle a={\tfrac multivariate gaussian distribution {1}{\sigma {\sqrt {2\pi }}}}} , b = μ, and c = σ. In mathematics, a Gaussian function,
Normal Distribution Pdf
often simply referred to as a Gaussian, is a function of the form: f ( x ) = a e − ( x − b ) 2 2
Normal Distribution Formula
c 2 {\displaystyle f\left(x\right)=ae^{-{\frac {(x-b)^{2}}{2c^{2}}}}} 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 normal distribution examples (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. Contents 1 Properties 2 Integral of a Gaussian function 2.1 Proof 3 Two-dimensional Gaussian function 3.1 Meaning of parameters for the general equation 4 Multi-dimensional Gaussian function 5 Gaussian profile estimation 6 Discrete Gaussian 7 Applications 8 See also 9 References 10 External links Properties[edit] 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. The parameter c is related to the full width at half maximum (FWHM) of the peak according to F W H M = 2 2 ln
peer An http://hyperphysics.phy-astr.gsu.edu/hbase/math/gaufcn.html error condition occurred while reading data from the network. Please retry your request. Your cache administrator is webmaster. Generated Sat, 15 Oct 2016 15:10:46 GMT by s_ac4 (squid/3.5.20)
the English-French mathematician de Moivre (in 1733), and later Laplace and Gauss. For the distribution corresponding to the curve we find names as: http://www.2dcurves.com/exponential/exponentialg.html de Moivre distribution Gaussian distribution Gauss Laplace distribution The Gaussian distribution can be https://www.probabilitycourse.com/chapter4/4_2_3_normal.php proved (by the so-called Central Limit Theorem) in the situation that each measurement is the result of a large amount of small, independent error sources. These errors have to be of the same magnitude, and as often positive as negative. When measuring a physical variable one tries to eliminate normal distribution systematic errors, so that only accidental errors have to be taken into account. In that case the measured values will spread around the average value, as a Gauss curve. It can be proved that in the case when the average value of a measured value is the 'best value', a Gaussian distribution holds. The 'best value' is here defined as that gaussian normal error value, for which the chance on subsequent measurements is maximal 1). Because in general an estimation of errors is rather rough, the distribution to be used has not to define the error very precise. More important is that the distribution is easy to work with. And the Gaussian distribution has that quality in many situations. Some real life examples of the Gauss distribution: distribution of the length of persons (given the sex) distribution of the weight of machine packed washing powder distribution of the diameter of machine made axes Taking the definition of the standard deviation 2) it can be seen that σ is the standard deviation in the Gauss distribution of the form: The points of inflection are situated at x = ± σ. For this distribution about two of the three measurements has a distance less than σ from the maximum value. And about one of the twenty measurements has a distance of more than 2σ. Another interesting quality of the Gauss curve is that it is the only function which remains unchanged for a Fourier transform. Because its form the
is by far the most important probability distribution. One of the main reasons for that is the Central Limit Theorem (CLT) that we will discuss later in the book. To give you an idea, the CLT states that if you add a large number of random variables, the distribution of the sum will be approximately normal under certain conditions. The importance of this result comes from the fact that many random variables in real life can be expressed as the sum of a large number of random variables and, by the CLT, we can argue that distribution of the sum should be normal. The CLT is one of the most important results in probability and we will discuss it later on. Here, we will introduce normal random variables. We first define the standard normal random variable. We will then see that we can obtain other normal random variables by scaling and shifting a standard normal random variable. A continuous random variable $Z$ is said to be a standard normal (standard Gaussian) random variable, shown as $Z \sim N(0,1)$, if its PDF is given by $$f_Z(z) = \frac{1}{\sqrt{2 \pi}} \exp\left\{-\frac{z^2}{2}\right\}, \hspace{20pt} \textrm{for all } z \in \mathbb{R}.$$ The $\frac{1}{\sqrt{2 \pi}}$ is there to make sure that the area under the PDF is equal to one. We will verify that this holds in the solved problems section. Figure 4.6 shows the PDF of the standard normal random variable. Fig.4.6 - PDF of the standard normal random variable. Let us find the mean and variance of the standard normal distribution. To do that, we will use a simple useful fact. Consider a function $g(u):\mathbb{R}\rightarrow\mathbb{R}$. If $g(u)$ is an odd function, i.e., $g(-u)=-g(u)$, and $|\int_{0}^{\infty} g(u) du| < \infty$, then $$\int_{-\infty}^{\infty} g(u) du=0.$$ For our purpose, let $$g(u)= u^{2k+1}\exp\left\{-\frac{u^2}{2}\right\},$$ where $k=0,1,2,...$. Then $g(u)$ is an odd function. Also $|\int_{0}^{\infty} g(u) du| < \infty$. One way to see this is to note that $g(u)$ decays faster t