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:

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