# Normal Distribution - Mathematics
Data Science for Political and Social Phenomena
Author: Guillaume Redoulès

# Normal Distribution

The probability density of normal distribution is:

$$\mathcal{N}(\mu,\sigma^2)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

where, * \(\mu\) is the mean (or expectation) of the distribution. It is also equal to median and mode of the distribution. * \(\sigma^2\) is the variance. * \(\sigma\) is the standard deviation.

## Standard Normal Distribution

If \(\mu=0\) and \(\sigma=1\), then the normal distribution is known as standard normal distribution:

$$\phi(x)=\frac{e^{-\frac{x^2}{2}}}{\sigma\sqrt{2\pi}}$$

Every normal distribution can be represented as standard normal distribution:

$$\mathcal{N}(\mu,\sigma^2)=\frac{1}{\sigma}\phi(\frac{x-\mu}{\sigma})$$

## Cumulative Probability

Consider a real-valued random variable, \(X\). The cumulative distribution function of \(X\) (or just the distribution function of \(X\)) evaluated at \(x\) is the probability that \(X\) will take a value less than or equal to \(x\):

$$F_X(x)=P(X\leq x)$$

also,

$$P(a\leq X\leq b)=P(a\lt X\lt b)=F_X(b)-F_X(a)$$

the cumulative distribution function for a function with normal distribution is:

$$\Phi(x)=\frac{1}{2}\left(1+erf\left(\frac{x-\mu}{\sigma\sqrt{2}}\right)\right)$$

where \(erf\) is the error function:

$$erf(z)=\frac{2}{\sqrt{\pi}}\int_0^ze^{-x^2}dx$$

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