Comments - A Beautiful Result in Probability Theory - Data Science Central2020-09-21T04:05:58Zhttps://www.datasciencecentral.com/profiles/comment/feed?attachedTo=6448529%3ABlogPost%3A827715&xn_auth=noJoe Blitzstein (teaching prob…tag:www.datasciencecentral.com,2019-05-20:6448529:Comment:8280392019-05-20T03:49:15.825ZVincent Granvillehttps://www.datasciencecentral.com/profile/VincentGranville
<p><a href="https://statistics.fas.harvard.edu/people/joseph-k-blitzstein" rel="noopener" target="_blank">Joe Blitzstein</a> (teaching probability at Harvard University) pointed out (see <a href="https://www.quora.com/What-is-the-variance-of-the-range-for-exponential-distributions" rel="noopener" target="_blank">here</a>) that my theorem is a particular case of a general result that applies to exponential distributions, known as the Renyi representation. This general result is illustrated in…</p>
<p><a href="https://statistics.fas.harvard.edu/people/joseph-k-blitzstein" target="_blank" rel="noopener">Joe Blitzstein</a> (teaching probability at Harvard University) pointed out (see <a href="https://www.quora.com/What-is-the-variance-of-the-range-for-exponential-distributions" target="_blank" rel="noopener">here</a>) that my theorem is a particular case of a general result that applies to exponential distributions, known as the Renyi representation. This general result is illustrated in the picture below and <a href="https://storage.ning.com/topology/rest/1.0/file/get/2647075955?profile=original" target="_blank" rel="noopener">in this document</a>.</p>
<p><a href="https://storage.ning.com/topology/rest/1.0/file/get/2647079173?profile=original" target="_blank" rel="noopener"><img src="https://storage.ning.com/topology/rest/1.0/file/get/2647079173?profile=RESIZE_710x" class="align-center"/></a></p>
<p><span>This also brings something very interesting: since my proof relies on the fact that the sum of the inverse of the squares is Pi^2/6 and since Renyi’s argument is entirely probabilistic, it is thus possible to prove, using probabilistic arguments alone, that the sum of the inverse of the squares is Pi^2/6. I will look at higher moments to see if there are some other facts about mathematical constants or integrals, that can be proved (thanks Renyi!) using probabilistic arguments alone. With some chance, I might even discover a new relationship.</span></p>
<p><span>Finally, another way to prove the result is to use the fact (see <a href="https://en.wikipedia.org/wiki/List_of_definite_integrals#Definite_integrals_involving_logarithmic_functions" target="_blank" rel="noopener">here</a>) that </span></p>
<p><span><a href="https://storage.ning.com/topology/rest/1.0/file/get/2647962026?profile=original" target="_blank" rel="noopener"><img src="https://storage.ning.com/topology/rest/1.0/file/get/2647962026?profile=RESIZE_710x" class="align-center"/></a></span></p>