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Topic: problem on record-breaking values in probability
Replies: 14   Last Post: Apr 14, 2013 11:36 PM

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David Bernier

Posts: 3,317
Registered: 12/13/04
problem on record-breaking values in probability
Posted: Feb 27, 2013 5:31 AM
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I used Marsaglia's 64-bit SUPER KISS pseudo-random number generator
to simulate uniform r.v.s on [0, 1] that are independent, as
X_1, X_2, X_3, ad infinitum

For each go, (or sequence) I define its 1st record-breaking value
as R(1) as X_1, its 2nd record-breaking value R(2) as the
value taken by X_n for the smallest n with X_n > X_1, and in general
R(k+1) as the value taken by the smallest n with X_n > R(k), for
k = 2, 3, 4, 5, ...

In my first simulation I get: R(20) = 0.999999999945556
or about 5.4E-11 less than 1 , a one in 18 billion event.

In fact, R(20) is about 1 - (0.307)^20 ...

So, I'm wondering about the asymptotics of 1 - R(k) for very
large k. Of course, R(k) is a andom variable with a
probability distribution. Can we say something about the
asymptotics of 1 - R(k) for large k?

David Bernier

--
dracut:/# lvm vgcfgrestore
File descriptor 9 (/.console_lock) leaked on lvm invocation. Parent PID
993: sh
Please specify a *single* volume group to restore.



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