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 . For this I would give 2 answers:
Substitution x2=t gives
And how is ?
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 converge to for (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!
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:
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.