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


Variance of Normal distribution

Variance of normal distribution is


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


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