# Questions tagged [error-function]

Use this tag for the error and complementary error functions (erf and erfc). These are special functions formed by taking definite integrals of the Gaussian/normal distribution function.

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### An upper bound involving the second derivative of the error function

I am trying to bound a function of the form \begin{align} f(x,y) &= \operatorname{erf}(x+y) - 2\operatorname{erf}(x) + \operatorname{erf}(x-y), \end{align} for small values of $y$ and all (...
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### What's $\int_{-\frac{\pi}{2}}^ {\frac{\pi}{2}}\text{erf}\left(\frac{\sqrt 2 R\cos\theta}{\sigma}\right)\text d\theta$?

The context of $$\int_{-\frac{\pi}{2}}^ {\frac{\pi}{2}}\text{erf}\left(\frac{\sqrt 2 R\cos\theta}{\sigma}\right)\text d\theta$$ is it came up whilst integrating the Rayleigh distribution function over ...
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### Solve or approximate the maxima of a function

Function is given by $$f(\lambda) = \lambda\exp\left(\frac{\lambda}{2}(\lambda a^2 - 2b)\right) \times \left(1 - \text{erf}\left(\frac{\lambda a^2 - b} {\sqrt{2} a}\right)\right)$$ where $\lambda>0$...
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### equation related to lagrange error formula.

So im trying to show that $E_2(x)$ = $\int_{a}^{x} \frac{f'''(t)(x-t)^2}{2}dt$. In the problem they want this to be done by using the equation $E_2(x)$ = $E_1(x)$ - $\frac{f''(a)(x-a)^2}{2}$ where we ...
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### Why does the quantile function of bivariate normal variables become non-elementary in one dimension?

I have been studying a bivariate random process where $X \sim N(0, \sigma_x), Y \sim N(0, \sigma_y)$. It turns out that finding an ellipse that covers proportion p of samples on this process is given ...
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### How to minimize the maximum absolute difference between 2 functions?: example $\min_a\{\|\text{erf}(x)-\tanh(\frac2{\sqrt{\pi}}(x+a x^3))\|_\infty\}$

How to minimize the maximum absolute difference between 2 functions?: example $\min_a\{\|\text{erf}(x)-\tanh(\frac2{\sqrt{\pi}}(x+a x^3))\|_\infty\}$ Intro_______________ In this other question I ...
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I am interested in a problem that involves computing the expectation of of the CDF $\Phi$ (or equivalently erfc shifted and scaled) for the standard normal distribution, for $x$ normal distributed ...