Circle Line Picking
المؤلف:
Sloane, N. J. A.
المصدر:
Sequences A000984/M1645, A001803/M2986, A061549, and A093581 in "The On-Line Encyclopedia of Integer Sequences."
الجزء والصفحة:
...
5-2-2020
1617
Circle Line Picking
Given a unit circle, pick two points at random on its circumference, forming a chord. Without loss of generality, the first point can be taken as 
, and the second by 
, with ![theta in [0,pi]](http://mathworld.wolfram.com/images/equations/CircleLinePicking/Inline3.gif)
(by symmetry, the range can be limited to 
instead of 
). The distance 
between the two points is then
 |
(1)
|
The average distance is then given by
 |
(2)
|
The probability density function 
is obtained from
 |
(3)
|
The raw moments are then
giving the first few as
(OEIS A000984 and OEIS A093581 and A001803), where the numerators of the odd terms are 4 times OEIS A061549.
The central moments are
giving the skewness and kurtosis excess as
Bertrand's problem asks for the probability that a chord drawn at random on a circle of radius 
has length 
.
REFERENCES:
Sloane, N. J. A. Sequences A000984/M1645, A001803/M2986, A061549, and A093581 in "The On-Line Encyclopedia of Integer Sequences."
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