How random is the digits of $\pi$?

Question

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Distribution of the digits of Pi

If we plot the digits of $\pi$

$3.1415926535897932384626433832795028841971693993751058\cdots$

versus the indices

$((1, 3), (2, 1), (3, 4), (4, 1), \cdots)$

How to quantify the randomness of the resultant curve?

The first part of the curve looks like this .

Rephrasing the question, suppose the curve happens to be a sample function of a random process, how to give the distribution of that random process?

In one quite important sense they are not random at all, because they are the one and only sequence that are the decimal digits of $\pi$. That's an extremely specific property. — hmakholm left over Monica, Dec 15 '12 at 16:30
There are lots of ways to try to quantify randomness. Which are appropriate will depend on what you intend to do with the result. — Chris Eagle, Dec 15 '12 at 16:35

akkkk · Answer 1 · 2012-12-15T17:06:31.470

4

Histogram of the first 10000 digits

Digits of Pi

Covariance of the first 10000 consecutive digits: -0.09 (negligible)

Mathematica code used:

Histogram:

Histogram[RealDigits[Pi, 10, 10000][[1]], 10]

Covariance: (I'm sure there's a more efficient way to do this)

N[Covariance[RealDigits[Pi, 10, 10000][[1]][[Range[1, 9999]]], 
   RealDigits[Pi, 10, 10000][[1]][[Range[2, 10000]]]]]

edited Dec 15 '12 at 17:06

answered Dec 15 '12 at 17:01

akkkk

1

@PeterSheldrick: mathematica: Histogram[RealDigits[Pi, 10, 10000][[1]], 10] – akkkk Dec 15 '12 at 17:02
If any of you guys have more requests, I'll be happy to compute/plot more data. – akkkk Dec 15 '12 at 17:04

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