Why does the area under the bell curve involve pi?

Question

Because we can obtain this curve by the simple procedure of flipping a coin, our gaussian/normal distribution curve/bell curve can be said to represent the law of averages.
If we accept this law as being ultimately inviolable, then one could argue that it is the most powerful force in the universe.
Because we encounter this mathematical form so often in nature, it seems surpising that it is defined by our two most important mathematical constants. Also f(sq.rt.2) = 1/e.
Thus pi, e and the law of averages, welded together in this way, can be said to represent a central and 'immortal' form.

e's role in this form is fairly clear, but why is pi there? The fact that pi arises in integrating f(x) we know, but is there a simple, logical reason for its presence?

Please, all suggestions welcome.

Answer 1

The main reason is that the bell curve has symmetry about a mean. Check out the following link that describes the derivation of the area under a bell curve (normal distribution) using an example of the origin as "bulls eye". The nature of pi involves 'symmetry': we start with the very fact that the circumference is completely symmetrical about the origin of any circle.

In very superficial terms, this is the pi-connection.