By Kendall M.G., Moran P.A.P.
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Extra resources for Geometrical Probability
Which, by calculating 9 the Jacobian, is seen to be nidi,. 21) i by itself therefore has the same distribution. This could also have been seen by observing that the n + 1 intervals are jointly distributed in the same way as the n+l intervals obtained by placing n+ 1 points at random on a circle of unit circumference. , Yk _, - Yk and Y^g,... . ^(\ we get _X \w fc for the joint distribution of of Ik+l s . 23) is obviously that serving the distribution of I v This is We can relate the distribution of n points in an interval to a Poisson process in two ways.
8 A more interesting situation arises when we have a fixed number of random points in an interval whose length we may take These as unity. , /n+1 Suppose that the interval is (0, 1) and that we have n random points whose coordiwill divide the interval into . X nates Then 19 ... 9 X . dXn (Q X^ ^ (0, 1). 19) 1). This can be regarded as a joint uniform distribution on an w-dimensional cube. ,Xn arranged in increasing order: X < Y < Y < Then the cube is ... dY*(Q < F l t is < ... 20) 1). then have = Y A 2T T 4,1 lt V J 2~ V -* 2 -*n l> Vn Vn = i-y.
Of sides (taken as equal to the number of segments of the great circles) and 24 respectively, whilst for general n it is 2n(n- 1). Thus, is 4, 12, 57 2 large the average area is asymptotically equal to 4jm~ , whilst the average number of sides of each quasi-polygonal region will as n becomes be the limit of which is 4. e. it is \nn, so that the l average perimeter of a region is asymptotically equal to ^nn~ and the -1 average length of each side is nn , . 4 tribution of regions in a small circle of radius r on the surface of the sphere.