I would be grateful for any ideas on the best approach to this problem:
Let x_1,x_2,...,x_n be n points iid uniformly in [0,1].
There are thus n-1 gaps g_i = x_{i+1} - x_i (i=1,2,...,n-1).
The cumulative distribution function of the kth gap is 1-(1-g)^n,
independent of k.
Now, if these gaps were independent RVs, we would know that the
density of the maximum gap was
rho(g) = n(n-1)(1-g)^(n-1)[1-(1-g)^n]^(n-2).
However, this is not correct because each g_i is correlated with its
neighbours.
Is there any way to fix this and get a closed formula for the density (or
cdf) for the
maximum gap?
Dr. Keith M. Briggs
Senior Mathematician, Complexity Research, BTexact Technologies
email: [log in to unmask]
phone: +44(0)1473 work: 641 911 home: 610 517 fax: 647 410
web: www.btexact.com/people/briggsk2/
mail: Keith Briggs, Antares 2pp5, Adastral Park, Martlesham, Suffolk
IP5 3RE, UK
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