Definition of Normal Distribution

A continuous random variable $X$ is said to have an normal distribution with parameter $\mu$ and $\sigma$ if its p.d.f. is given by $$ \begin{equation*} f(x)=\left\{ \begin{array}{ll} \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\big(\frac{x-\mu}{\sigma}\big)^2}, & \hbox{$-\infty< x<\infty ;-\infty < \mu < \infty; \sigma^2>0$;} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{equation*} $$

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