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From: ags@pucc-i (Seaman)
Newsgroups: net.math,net.puzzle
Subject: Chain problem - clarification(?)
Message-ID: <222@pucc-i>
Date: Sat, 25-Feb-84 10:37:54 EST
Article-I.D.: pucc-i.222
Posted: Sat Feb 25 10:37:54 1984
Date-Received: Mon, 27-Feb-84 04:19:50 EST
Organization: Purdue University Computing Center
Lines: 45

How long are the required lengths of chain?  Let's take an analogy:
Ask someone to choose a positive real number "at random."  What is
the probability that the chosen number is

	(1) less than 1?
	(2) less than 10?
	(3) less than 100?
	(4) less than 1000?

It's rather difficult to assign these probabilities in any objective fashion.
One thing does seem reasonable, though:  the probability density decreases
as the numbers get large.

Consider these three events:

	(a) The number is between 3 and 4.
	(b) The number is between 1003 and 1004.
	(c) The number is between 1,000,003 and 1,000,004.

Event (a) seems more likely than (b), which seems more likely than (c).
The probability of (c), though small, is certainly not zero.

I suggest that a reasonable type of distribution to use in this problem is the 
exponential destribution, given by the density function

	p(t) = exp(-a * t)	for t >= 0.

Notice that the integral (from 0 to inf.) of p(t) dt = 1, so that we do have
a probability density.  The constant a turns out (by a simple integration) to
be the EXPECTED VALUE of the distribution.

Let us make the chain problem more precise:

	The required lengths are x and y, which we assume are independent
	random variables, exponentially distributed with expected value a.

If anyone has a different distribution that seems likely, let's hear that
one also.
-- 

Dave Seaman
..!pur-ee!pucc-i:ags

"Against people who give vent to their loquacity 
by extraneous bombastic circumlocution."