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--- un:gaussian-distribution [2023/06/20 20:21] – asad
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-====== Gaussian distribution ======
-Also known as the normal distribution or bell curve, is a continuous [[probability distribution]] that is widely used in statistics, mathematics, and the natural and social sciences. It is named after the German mathematician [[Gauss]].
-The Gaussian distribution is characterized by its symmetric bell-shaped curve, which is defined by two parameters: the mean ( $\mu$ ) and the standard deviation ( $\sigma$ ). The mean represents the center of the distribution, while the standard deviation determines the spread or dispersion of the data.
-If a population has a normal distribution, then in a single trial the probability of getting an outcome  $x$  is
- $P(x,\mu,\sigma) = \frac{1}{\sigma\sqrt{2\pi}} e^{[(x-\mu)/\sigma]^2/2}$
-where  $e$  is [[Euler's number]]. The distribution is shown below for three different mean and standard deviations.
-[[https://colab.research.google.com/drive/1S9shHapVuaLUiuPY84EA_mDYK6JOZVCF?usp=sharing|{{:un:gaussian.png?nolink&700|}}]]
-The full width at half-maximum  $\text{FWHM} = 2.354\sigma$  and probable error  $\text{PE}=0.6745\sigma=0.2865\text{ FWHM}$ . The standard deviation, FWHM and PE are shown below.
-[[https://colab.research.google.com/drive/1S9shHapVuaLUiuPY84EA_mDYK6JOZVCF?usp=sharing|{{:un:normal.png?nolink&700|}}]]