> [!NOTE] Definition (Standard Normal Distribution) > The standard normal distribution is the [[Normal Distribution|normal distribution]] with parameters $0$ and $1:$ that is, the [[Probability Density Function|probability density function]] is given by $f_{X}(x)=\frac{1}{\sqrt{2\pi}}\cdot e^{-\frac{x^2}2}.$ # Properties By [[Expectation of Standard Normal Distribution]], if $X\sim\mathcal{N}(0,1)$ then $\mathbb{E}[X]=0.$ By [[Variance of Standard Normal Distribution]], if $X\sim\mathcal{N}(0,1)$ then $\text{Var}(X)=1.$ # Applications By [[Standard Normal Random Variable as Transformation of Normal Random Variable]], if $X\sim \mathcal{N}(\mu,\sigma^{2})$ then $\frac{X-\mu}{\sigma}\sim\mathcal{N}(0,1).$