> [!NOTE] Definition (Normal Distribution) >Let $X$ be a [[Continuous random variables|continuous real-valued random variable]]. Let $\mu\in \mathbb{R}$ and $\sigma>0.$ Then $X$ has a normal (or Gaussian) distribution with parameters $\mu$ and $\sigma^{2}$, denoted $X\sim\mathcal{N}(\mu,\sigma^{2}),$ if its [[Probability Density Function|probability density function]] is given by $f_{X}(x)=\frac{1}{\sqrt{ 2 \pi \sigma^{2} }} \cdot e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}}$ **Note**: By [[Normal Distribution Probability Density Function is Probability Density Function]], $\int_{-\infty}^{\infty} f_{X}(x) \, dx$ is indeed $1.$ # Properties By [[Standard Normal Random Variable as Transformation of Normal Random Variable]], if $X\sim \mathcal{N}(\mu,\sigma^{2})$ then $\frac{X-\mu}{\sigma}\sim\mathcal{N}(0,1),$ where $\mathcal{N}(0,1)$ is known as the [[Standard Normal Distribution|standard normal distribution]]. By [[Expectation of Normal Distribution]], .... By [[Variance of Normal Distribution]], ...