## 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