What is the Gaussian function?

Hint: The Gaussian function is mainly used to solve normal distribution problems. It is also known as the probability density function (PDF) of normal distribution.
In this, we will see how the Gaussian function is made and in what fields it is used.
The terms that we will be using in this solution are:
Mean $\overline x $ - is the average value of any data.
Variance ${\sigma ^2}$ - simply shows how much a random variable differs from the expected value.


Complete step by step answer:
The basic Gaussian function is given by $y = {e^{ - {x^2}}}$ .
Now, we will parameterize it with some constants, then it becomes, $y = A{e^{ - b{{(x - c)}^2}}}$ .
Where we will define the constants $A$ , $b$ and $c$ , when we have to use it for statistical purposes, i.e., when we want to make it into a standard normal distribution.
Then, $c$ becomes the mean, $b$ becomes the half of the reciprocal of the variance and we choose $A$ in such a way that the integral of the function overall $x$ is $1$ .
That is, $c = \mu $ , $b = \dfrac{1}{{2{\sigma ^2}}}$ and $A = \dfrac{1}{{\sqrt {2\pi {\sigma ^2}} }}$ .
Then, the normal probability distribution function or the gaussian function is given $f\left( {x|\mu ,{\sigma ^2}} \right) = \dfrac{1}{{\sqrt {2\pi {\sigma ^2}} }}{e^{ - \dfrac{{{{\left( {x - \mu } \right)}^2}}}{{2{\sigma ^2}}}}}$ .

Note: Gaussian function is most important in probability distribution as it fits many phenomena like age, height, the sum of rolls of two dice, test scores, and so on.
The Gaussian function is used in statistics to describe the various methods that are normal distributions method and in mathematics to solve heat equations method, diffusion equation method and to define Weierstrass transformation.
It is also used in error analysis, to determine the significance of measurement, as it is a well-approximated method for the noise present in the distribution function of a random set.