# The Adventure of π to the Normal distribution

Today, after giving the definition of the normal distribution a student asked me a very interesting question: how does the number π emerge in the Gaussian curve? The question is legitimate, because the Gaussian curve does not show any relation to the circle curve. I improvised an brief answer, and I will try to give here the path of this fascinating adventure of π. Firstly, the distribution function of a normal distributed random variable with expectation μ and variance σ2 is

and its limit is

This is a normalised form of the Poisson integral:

Now the question is why this equals $\sqrt{\pi}$. For this I would give 2 answers:

### 1. Path

Substitution x2=t gives

And how is $\Gamma \left(\frac 1 2\right) = \sqrt{\pi}$?

This can be explained over the Gauss form of the Gamma function:

These 2 different forms give at ½ when multiplied with each other the double of the Wallis product:

Now, why does $W(n):=\frac{k^2}{k^2 - \frac 1 4} \prod_{k=1}^n$ converge to $\frac \pi 2$ for $n\rightarrow\infty$ (Wallis product)? This is because of the recursive relation

The recursive resolution of the terms of the following fraction yields:

and therefore the equation of the Wallis product.

We see here that π comes from the sine function to the approximation of the Wallis product, then as the value of Γ(½) and from there it evaluates the square of the Poisson integral. A long path!

### 2. Path

The second path that I will present is shorter and more direct. Because

calculating the left side should give the expected result (π). This can be done by integrating over a circle with centre in 0 and radius a (B(a)) and then by transformation to polar coordinates:

### References:

Forster Otto: Analysis 1; Differential- und Integralrechnung einer Veränderlichen, 6,. verbesserte Auflage, o.O., vieweg, 2001.

Κουνιας Στρατής, χρόνης Μωυσιάδης: Θεωρία Πιθανοτήτων Ι, Κλασική πιθανότητα, μονοδιάστατες κατανομές, Θεσσαλονίκη, Εκδόσης Ζήτη, 1999.

PS. Wikipedia has an interesting article about the calculation of the Gaussian integral.