Normal Distribution : A new perspective

This article tries to look at the Normal Distribution in somewhat of a different angle. Most of us have seen the probability density function and the famous bell shaped curve very often . So through this article I want you to gain a different insight about the same.



If you have a background in mathematics or statistics , you must be familiar with the following bell shaped curve and the somewhat difficult looking and inherently less intuitive function :P

If not , let me introduce them as the Normal distribution and its Probability Density Function.




                  

Let's try and get familiarized with both of these :)

We consider a simpler case , where μ = 0 and σ = 1, that is the case of Standard Normal Distribution. However , all the things apply to normal distribution also.


So , we have :






Let's try and break down the the function (probability density function) to gain a new perspective.




Let's try and make a connection about the function and the bell shape of the curve.




Step 1 : Start with plotting g(x) = x²







This is a very familiar parabola.



Step 2 : Now let's take it a step forward and plot h(x) = x² / 2





This is just another version of a parabola we had initially .


Step 3 : Now let's do a transformation and plot k(x) =









Step 4 : Now let's take the transformation which takes the reciprocal of the previous function:

i(x) =


Boom !!

The python code for generating these graphs is attached on my google drive on the following link :

https://drive.google.com/file/d/1Fm2zljQp0og01ytD1xOd5yR07RfvYzbN/view?usp=sharing

So , here we are with a function that looks like our bell shaped curve.The next thing we need to answer is about the constant , that is :



   multiplied to the function i(x) .


Note , after multiplying i(x) by the constant , we get the standard normal density function.
But we need to answer the question , why do we do this....

Spoiler Alert!! :P

We do that to make the function a valid Probability Density Function.




Mathematically Speaking :

We know that for any function to be a valid density function, it should follow the following rules :



1. i(x) ≥ 0 for all x.

2.The total area under the curve should equal to 1.



Clearly , property no. 1 is satisfied as we can clearly see the plot is in the positive region.


Let's check the if the second condition holds or not .



For that we need to try and solve for i(x) integrated over the complete domain.


The technique used here was given by Pierre-Simon Laplace who was a french scholar with wide contributions in Statistics , Mathematics , Physics  . He gave this technique in the early nineteenth century(1812 precisely).


Consider solving the integral :




We take a simpler version of it to begin with :

Since is an even function , that is :


f(-x) = f(x) for all x


So we can consider only the positive part of x axis and then twice that integral as the f(x) is same for the positive as well as negative x values.



We compute :






Now we since directly solving the integral is not possible , so this was a way to go about solving the integral .




Assume :

Note that surprisingly the integral becomes easier to solve if we have one extra term in the integral .

Hence :

More Generally ,

In our case we have a = 1/2 and b = 0 , So finally




So , to make i(x) a valid Probability Distribution Function , we multiply i(x) by


This finally gives us the standard normal density Function.

I hope this article gave you some different perspective about the normal distribution.

If you liked this article , do comment and share .

Happy Learning!

Thanks

Sahaj Thareja