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A Derivation of Wallis's Formula for Pi

For \(\pi\) day plus one, here is a derivation of Wallis's formula for \(\pi\) that is based on probability distributions. Start with the binomial distribution for a fair coin toss. The probability of getting \(k\) heads on \(n\) tosses is


Assume an even number of tosses, \(n=2m\). Then the distribution has a maximum at \(k=m\) of


For large \(n\), the binomial distribution can be closely approximated by a Normal probability density function give by


where \(\mu=n/2=m\) and \(\sigma^2=n/4=m/2\). The maximum at \(x=\mu\) is


Comparing this with the maximum for the binomial distribution and taking the limit as \(m\to\infty\), you get


Now we just need the following double factorial identities



Then we can write


and \(\pi\) can be expressed as


or in product form as


This is almost Wallis's formula. To get there, note that


Substitute this into the above formula, take the limit and you get Wallis's formula


Is that cool or what?

© 2010-2016 Stefan Hollos and Richard Hollos

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