ags@pucc-i (Seaman) (02/25/84)
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."
jab@nwuxd.UUCP (jab) (02/27/84)
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. --- Let's see. First, let's through out (2), (3), and (4), since WLOG we can use the same argument as we'll use for (1) --- just scale the "random number". Now, since there are an infinite number of intervals (n, n+1], in the range (0, infinity), the odds of you fixing "n" and then picking a "random" positive real number in the interval (n, n+1] is almost zero. Make sense? Let's fix "n" and try it. I'll pick a "random" positive real number. Since there are exactly as many real numbers in the range (n, n+1] as ((0, n] union (n+1, infinity)), there should be a 1 in 2 chance I pick a number in the range (n, n+1]. Right? What's wrong? I just decided earlier that it should be almost zero, now it looks like it should be 1 in 2. Since we're working with something that isn't finite, we seem to have wandered into an indeterminate. I don't think that it's possible to express what you're asking for in terms of conventional probabilities --- there's too many intervals to play with. It's like the friend who commented that "choosing a random integer is dumb, since the odds of you picking one that you can pronounce in your lifetime are almost zero." Jeff Bowles Lisle, IL
markb@sdcrdcf.UUCP (Mark Biggar) (02/28/84)
Why are all of you worried about chooseing random real numbers?
A chain can only have lengths that are an integral multiple of a length
some what less then the length of one link. Chains are not produceable
in lengths of other than 1 link, 2 links, etc.
Now does this amke any diffenence in the problem?
Mark Biggar
{allegra,ihnp4,sdcsvax,cbosgd,trw-unix}!sdcrdcf!markb