2
$\begingroup$

Possible Duplicate:
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 .

Re: @Henning Makholm

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?

  • 10
    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.2012-12-15
  • 2
    You might like to take a look at [Normal Numbers](http://en.wikipedia.org/wiki/Normal_number).2012-12-15
  • 1
    There are lots of ways to try to quantify randomness. Which are appropriate will depend on what you intend to do with the result.2012-12-15

1 Answers 1

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]]]]]
  • 1
    @PeterSheldrick: mathematica: Histogram[RealDigits[Pi, 10, 10000][[1]], 10]2012-12-15
  • 0
    If any of you guys have more requests, I'll be happy to compute/plot more data.2012-12-15