Gaussian Distribution

A Gaussian or normal distribution always has a bell-shaped curve and is determined by two parameters: the mean $\mu$ and the standard deviation $\sigma$. The graph associated with this phenomenon is symmetric with its center at the mean, and its shape is determined by the standard deviation. The value of this function at a point $x$ is given by formula

$$p(x)=\frac{1}{\sqrt{2\pi {\sigma }^2}}\exp \left(-\frac{(x-\mu {)}^2}{2{\sigma }^2}\right).$$

The term $\frac{1}{\sqrt{2\pi {\sigma }^2}}$ ensures that the integral of the function over the entire range of $x$ converges to 1.

$$\int p(x)dx=1$$