Normal Error Probability Function
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that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as:[1][2] erf ( x ) = 1 π ∫ − x x e − erf function t 2 d t = 2 π ∫ 0 x e − t
Normal Distribution Equation
2 d t . {\displaystyle {\begin − 6\operatorname − 5 (x)&={\frac − 4{\sqrt {\pi }}}\int _{-x}^ − 3e^{-t^ − normal distribution formula 2}\,\mathrm − 1 t\\&={\frac − 0{\sqrt {\pi }}}\int _ 9^ 8e^{-t^ 7}\,\mathrm 6 t.\end 5}} The complementary error function, denoted erfc, is defined as erfc normal distribution examples ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − t 2 d t = e − x 2 erfcx ( x ) , {\displaystyle {\begin 2\operatorname 1 (x)&=1-\operatorname 0 (x)\\&={\frac Φ 9{\sqrt {\pi }}}\int _ Φ 8^{\infty }e^{-t^ Φ 7}\,\mathrm Φ 6 t\\&=e^{-x^ Φ 5}\operatorname Φ 4 (x),\end Φ 3}}
Normal Distribution Pdf
which also defines erfcx, the scaled complementary error function[3] (which can be used instead of erfc to avoid arithmetic underflow[3][4]). Another form of erfc ( x ) {\displaystyle \operatorname 2 (x)} for non-negative x {\displaystyle x} is known as Craig's formula:[5] erfc ( x | x ≥ 0 ) = 2 π ∫ 0 π / 2 exp ( − x 2 sin 2 θ ) d θ . {\displaystyle \operatorname 0 (x|x\geq 0)={\frac Φ 9{\pi }}\int _ Φ 8^{\pi /2}\exp \left(-{\frac Φ 7}{\sin ^ Φ 6\theta }}\right)d\theta \,.} The imaginary error function, denoted erfi, is defined as erfi ( x ) = − i erf ( i x ) = 2 π ∫ 0 x e t 2 d t = 2 π e x 2 D ( x ) , {\displaystyle {\begin Φ 0\operatorname − 9 (x)&=-i\operatorname − 8 (ix)\\&={\frac − 7{\sqrt {\pi }}}\int _ − 6^ − 5e^ − 4}\,\mathrm − 3 t\\&={\frac − 2{\sqrt {\pi }}}e^ − 1}D(x),\end − 0}} where D(x) is the Dawson function (which can be used instead of erfi to avoid
of the normal distribution is \( f(x) = \frac{e^{-(x - \mu)^{2}/(2\sigma^{2}) }} {\sigma\sqrt{2\pi}} \) where μ is the location parameter and σ is the scale parameter. The case where μ = 0 and σ = 1 is normal distribution statistics called the standard normal distribution. The equation for the standard normal distribution is normal distribution standard deviation \( f(x) = \frac{e^{-x^{2}/2}} {\sqrt{2\pi}} \) Since the general form of probability functions can be expressed in terms of the
Multivariate Gaussian Distribution
standard distribution, all subsequent formulas in this section are given for the standard form of the function. The following is the plot of the standard normal probability density function. Cumulative Distribution Function https://en.wikipedia.org/wiki/Error_function The formula for the cumulative distribution function of the standard normal distribution is \( F(x) = \int_{-\infty}^{x} \frac{e^{-x^{2}/2}} {\sqrt{2\pi}} \) Note that this integral does not exist in a simple closed formula. It is computed numerically. The following is the plot of the normal cumulative distribution function. Percent Point Function The formula for the percent point function of the normal distribution does not exist in a http://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm simple closed formula. It is computed numerically. The following is the plot of the normal percent point function. Hazard Function The formula for the hazard function of the normal distribution is \( h(x) = \frac{\phi(x)} {\Phi(-x)} \) where \(\phi\) is the cumulative distribution function of the standard normal distribution and Φ is the probability density function of the standard normal distribution. The following is the plot of the normal hazard function. Cumulative Hazard Function The normal cumulative hazard function can be computed from the normal cumulative distribution function. The following is the plot of the normal cumulative hazard function. Survival Function The normal survival function can be computed from the normal cumulative distribution function. The following is the plot of the normal survival function. Inverse Survival Function The normal inverse survival function can be computed from the normal percent point function. The following is the plot of the normal inverse survival function. Common Statistics Mean The location parameter μ. Median The location parameter μ. Mode The location parameter μ. Range \(-\infty\) to \(\infty\). Standard Deviation The scale parameter σ. Coefficient of Variation σ/μ Skewness 0 Kurtosis 3 Parameter Estimation The location and scale para
here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings http://math.stackexchange.com/questions/37889/why-is-the-error-function-defined-as-it-is and policies of this site About Us Learn more about Stack Overflow the company Business Learn more about hiring developers or posting ads with us Mathematics Questions Tags Users Badges Unanswered Ask Question _ Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it normal distribution only takes a minute: Sign up Here's how it works: Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Why is the error function defined as it is? up vote 35 down vote favorite 6 $\newcommand{\erf}{\operatorname{erf}}$ This may be a very naïve question, but here goes. The error function normal error probability $\erf$ is defined by $$\erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2}dt.$$ Of course, it is closely related to the normal cdf $$\Phi(x) = P(N < x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-t^2/2}dt$$ (where $N \sim N(0,1)$ is a standard normal) by the expression $\erf(x) = 2\Phi(x \sqrt{2})-1$. My question is: Why is it natural or useful to define $\erf$ normalized in this way? I may be biased: as a probabilist, I think much more naturally in terms of $\Phi$. However, anytime I want to compute something, I find that my calculator or math library only provides $\erf$, and I have to go check a textbook or Wikipedia to remember where all the $1$s and $2$s go. Being charitable, I have to assume that $\erf$ was invented for some reason other than to cause me annoyance, so I would like to know what it is. If nothing else, it might help me remember the definition. Wikipedia says: The standard normal cdf is used more often in probability and statistics, and the error function is used more often in other
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