Normal Distribution
A continuous random variable $X$ is said to have a normal
distribution if its probability density function is given by
$$ \begin{equation*} f(x;\mu, \sigma^2) = \left\{ \begin{array}{ll} \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2\sigma^2}(x-\mu)^2}, & \hbox{$-\infty< x<\infty$, $-\infty<\mu<\infty$, $\sigma^2>0$;} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{equation*} $$
where $\mu$ and $\sigma^2$ are the parameters of normal
distribution. In notation it can be written as $X\sim N(\mu,
\sigma^2)$.
Mean Normal distribution
The mean of normal distribution is
$E(X)=\mu$.
Variance of Normal distribution
Variance of normal distribution is
$V(X)=\sigma^2$.
Mode of Normal distribution
The mode of normal distribution is
Mode = $\mu$.
Median of Normal distribution
The median of normal distribution is
Median = $\mu$.
M.G.F. of Normal Distribution
The m.g.f. of $N(\mu,\sigma^2)$ distribution is
$M_X(t)=e^{t\mu +\frac{1}{2}t^2\sigma^2}$
Mean deviation about mean
The mean deviation about mean is