Normal distribution

In probability theory and statistics, the normal distribution (sometimes Gaussian distribution, bell curve) is a probability distribution given by the pdf:

$$f(x)=\frac{1}{\sigma \sqrt{2\pi }}\exp (-\frac{(x-\mu {)}^2}{2{\sigma }^2})$$ $$X\sim N(\mu ,{\sigma }^2)$$

where $\mu$ is the mean and ${\sigma }^2$ is the variance. Adjusting the mean and variances scales the shape of the distribution. The unit CDF ($\mu =0,\sigma =1$) is:

$$F_X(x)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^x\exp \left(\frac{t^2}{2}\right)dt$$

And the CDF is given by:

$$F_X(x)=\frac{1}{\sigma \sqrt{2\pi }}{\int }_{-\infty }^x\exp \left(-\frac{(x-\mu {)}^2}{2{\sigma }^2}\right)dx$$

Other relevant formulas

The $\sigma$ parameter can be produced with the softplus function:

$$\zeta (x)=\log (1+e^x)$$

The characteristic function for Gaussian distributions are given by:

$${\Phi }_X(\omega )=\exp \left(j\omega \mu -\frac{{\omega }^2{\sigma }^2}{2}\right)$$

Properties

One of the key properties of the normal distribution is that:

• About 68% of the population lies within one standard deviation of the median.
• About 95% within two and 99.7% within three.

Because this idea is true for any normal population, we may be motivated to convert into a set of “standard units”. This leads to the idea of the z-score, which we can use to work backwards and calculate the mean/standard deviation/sample.