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Introduction

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Whether or not you are familiar with statistics, you probably have heard of the so-called normal -distribution or bell curve from a variety of places such as finance, biology, physics, or the study -of polling. We may also encounter it in interesting facts like - -the human height is normally distributed. And school teachers would often assume -that the distribution of test scores is a bell curve. In words, the normal distribution shows -that data near the average value are more likely to occur than those far from the average. -Mathematically, the distribution (or the probability density function) is

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\begin{equation} -f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac 1 2 \left(\frac{x-\mu}{\sigma}\right)^2}. -\end{equation}

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Here $\mu$ is the average of the distribution and the parameter $\sigma$ is its standard deviation. -If we plot the function $f(x)$ for zero mean $\mu=0$ and unit standard deviation $\sigma=1$, -the curve looks like the following

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Normal Distribution

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Hence the nickname bell curve.

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In mathematics, the Gaussian integral is the integration of the Gaussian function -$f(x) = e^{-x^2}$. The integral is

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\begin{equation} -\int_{-\infty}^\infty e^{-x^2},dx = \sqrt{\pi}. -\end{equation}

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This integral is the basis of the calculation of the normalization constant

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\begin{equation} -\frac{e^{-z^2/2}}{1-e^{-\sqrt{\pi}(1+i)z}} -\end{equation}

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