Comments - 88 percent of all integers have a factor under 100 - Data Science Central2021-07-25T10:25:27Zhttps://www.datasciencecentral.com/profiles/comment/feed?attachedTo=6448529%3ABlogPost%3A545883&xn_auth=noFor those wondering how you c…tag:www.datasciencecentral.com,2017-04-03:6448529:Comment:5465202017-04-03T21:35:33.320ZVincent Granvillehttps://www.datasciencecentral.com/profile/VincentGranville
<p>For those wondering how you can even put a distribution on the infinite set of integers, here is the answer: We are dealing with a <a href="http://mathworld.wolfram.com/MarkedPointProcess.html" target="_blank">marked point process</a>, in which the points are the positive integers, and the mark is 1 if a point is divisible by a number less than 100, 0 otherwise. It is then possible to compute the probability for the mark to be either 0 or 1. This is one of the basic distributions associated…</p>
<p>For those wondering how you can even put a distribution on the infinite set of integers, here is the answer: We are dealing with a <a href="http://mathworld.wolfram.com/MarkedPointProcess.html" target="_blank">marked point process</a>, in which the points are the positive integers, and the mark is 1 if a point is divisible by a number less than 100, 0 otherwise. It is then possible to compute the probability for the mark to be either 0 or 1. This is one of the basic distributions associated with a marked point process. </p>
<p>People may not notice it because in this case, the points (integers) are equally spaced while in most point processes, the points are randomly distributed according to some particular distribution. Only the marks appear as somewhat random here. Note that the number of points is also infinite in most (but not all) examples of point processes. <span>These processes are sometimes called doubly-stochastic in the sense that randomness can occur for the location of the points, for the marks, or both. Here randomness occurs only for the marks. </span>Another way to look at it: Let's consider the points to be truly randomly distributed, with the inter-point distances having an exponential distribution with variance (say) equal to <em>v</em>, as in a Poisson process. Let <em>v</em> tends to zero, then you get points that are equally spaced.</p>
<p>You may also want to read my article on <a href="http://www.datasciencecentral.com/profiles/blogs/prime-numbers-interesting-distribution-and-density-results" target="_blank">how to define and measure integer densities</a>, with a different, non-probabilistic perspective.</p>