Normal Distribution (Gaussian Distribution)

Standard Normal Distribution

A continuous r.v. $Z$ is said to have the standard normal distribution if its PDF $\phi$ is given by

$$\phi (z)=\frac{1}{\sqrt{2\pi }}{e}^{-z^2/2},-\infty <z<\infty$$

We write this as $Z\sim \mathcal{N}(0,1)$ since, as we could obtain $Z$ has mean $0$ and variance $1$

Normal Distribution

If $Z\sim \mathcal{N}(0,1)$, then

$$X=\mu +\sigma Z$$

is said to have the normal distribution with mean $\mu$ and variance ${\sigma }^2$, for any real $\mu$ and ${\sigma }^2$ with $\sigma >0$. We denote this by $X\sim \mathcal{N}(\mu ,{\sigma }^2)$