# Gaussian Distribution Formula

Gaussian distribution is very common in a continuous probability distribution. The Gaussian distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables. Check out the Gaussian distribution formula below.

## Formula of Gaussian Distribution

The probability density function formula for Gaussian distribution is given by,

$\large f(x,\mu , \sigma )=\frac{1}{\sigma \sqrt{2\pi}}\; e^{\frac{-(x- \mu)^{2}}{2\sigma ^{2}}}$

Where,

$$\begin{array}{l}x\end{array}$$
is the variable

$$\begin{array}{l}\mu\end{array}$$
is the mean

$$\begin{array}{l}\sigma\end{array}$$
is the standard deviation

### Solved Examples

Question 1: Calculate the probability density function of Gaussian distribution using the following data. x = 2,

$$\begin{array}{l}\mu\end{array}$$
= 5 and
$$\begin{array}{l}\sigma\end{array}$$
= 3

Solution:

From the question it is given that, x = 2,

$$\begin{array}{l}\mu\end{array}$$
= 5 and
$$\begin{array}{l}\sigma\end{array}$$
= 3

Probability density function formula of Gaussian distribution is,

f(x,

$$\begin{array}{l}\mu\end{array}$$
,
$$\begin{array}{l}\sigma\end{array}$$
) =
$$\begin{array}{l}\frac{1}{\sigma \sqrt{2\pi }}\end{array}$$
$$\begin{array}{l}\;\end{array}$$
$$\begin{array}{l}e^{\frac{-(x-\mu )^{2}}{2\sigma ^{2}}}\end{array}$$

f(2, 5, 3 ) =

$$\begin{array}{l}\frac{1}{\sigma \sqrt{2\pi }}\end{array}$$
$$\begin{array}{l}e^{\frac{-(2-5 )^{2}}{18}}\end{array}$$

=[1/(3 ×2.51) ] × 0.6065

= 0.1328 ×0.6065

= 0.0805