Normal Distribution

The normal distribution is also called Gaussian distribution.

For a random variable $\it{X}$ with the expected value $\mu_X$ and variance $\sigma_X^2$, the probability density function is:

$$f(\it{X}) = \frac{1}{\sigma_X \sqrt{2\pi}} e^{-\frac{1}{2}\big(\frac{\it{X} - \mu_X}{\sigma_X}\big)^2} \quad \text{where } -\infty < \it{X} < \infty$$

It is written as:

$$\it{X} \sim \it{N}(\mu_X, \sigma_X^2)$$

$\it{Z}$ is a standard normal variable:

$$\it{Z} = \frac{X - \mu_X}{\sigma_X}$$

Therefore, the standard normal distribution is a special case for the normal distribution:

$$\it{Z} \sim \it{N}(0, 1)$$

$$f(\it{Z}) = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}\it{Z}^2}$$