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,

$x$ is the variable

$\mu$ is the mean

$\sigma$ is the standard deviation

### Solved Examples

**Question 1: **Calculate the probability density function of Gaussian distribution using the following data. x = 2, $\mu$ = 5 and $\sigma$ = 3

**Solution:**

From the question it is given that, x = 2, $\mu$ = 5 and $\sigma$ = 3

Probability density function formula of Gaussian distribution is,

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

f(2, 5, 3 ) = $\frac{1}{\sigma \sqrt{2\pi }}$$e^{\frac{-(2-5 )^{2}}{18}}$

=[1/(3 ×2.51) ] × 0.6065

= 0.1328 ×0.6065

= 0.0805