For π day plus one, here is a derivation of Wallis's formula for π 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
(nk)12n
Assume an even number of tosses, n=2m. Then the distribution has a maximum at k=m of
(2mm)122m
For large n, the binomial distribution can be closely approximated by a Normal probability density function give by
p(x)=1σ√2πexp−(x−μ)22σ2
where μ=n/2=m and σ2=n/4=m/2. The maximum at x=μ is
p(μ)=1√mπ
Comparing this with the maximum for the binomial distribution and taking the limit as m→∞, you get
1√π=limm→∞(2mm)√m22m
Now we just need the following double factorial identities
(2m)!=(2m−1)!!2mm!
2mm!=(2m)!!
Then we can write
(2mm)√m22m=(2m−1)!!(2m)!!√m
and π can be expressed as
π=limm→∞1m((2m)!!(2m−1)!!)2
or in product form as
π=limm→∞1mm∏k=1(2k2k−1)2
This is almost Wallis's formula. To get there, note that
m∏k=1(2k−1)2=12m+1m∏k=1(2k−1)(2k+1)
Substitute this into the above formula, take the limit and you get Wallis's formula
π=2∞∏k=1(2k)2(2k−1)(2k+1)
Is that cool or what?
© 2010-2016 Stefan Hollos and Richard Hollos
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