{"id":9956,"date":"2020-05-05T11:19:54","date_gmt":"2020-05-05T15:19:54","guid":{"rendered":"https:\/\/spinor.info\/weblog\/?p=9956"},"modified":"2020-05-05T16:27:46","modified_gmt":"2020-05-05T20:27:46","slug":"some-probability-distributions","status":"publish","type":"post","link":"https:\/\/spinor.info\/weblog\/?p=9956","title":{"rendered":"Some probability distributions"},"content":{"rendered":"

So here I am, reading about some trivial yet not-so-trivial probability distributions.<\/p>\n

Let’s start with the uniform distribution. Easy-peasy, isn’t it: a random number, between 0 and 1, with an equal probability assigned to any value within this range.<\/p>\n

So… what happens if I take two such random numbers and add them? Why, I get a random number between 0 and 2 of course. But the probability distribution will no longer be uniform. There are more ways to get a value in the vicinity of 1 than near 0 or 2.<\/p>\n

And what happens if I add three such random numbers? Or four? And so on?<\/p>\n

The statistics of this result are captured by the Irwin-Hall distribution<\/a>, defined as<\/p>\n

$$f_{\\rm IH}(x,n)=\\dfrac{1}{2(n-1)!}\\sum\\limits_{k=1}^n(-1)^k\\begin{pmatrix}n\\\\k\\end{pmatrix}(x-k)^{n-1}{\\rm sgn}(x-k).$$<\/p>\n

\"\"<\/a><\/p>\n

OK, so that’s what happens when we add these uniformly generated random values. What happens when we average them? This, in turn, is captured by the Bates distribution<\/a>, which, unsurprisingly, is just the Irwin-Hall distribution, scaled by the factor \\(n\\):<\/p>\n

$$f_{\\rm B}(x,n)=\\dfrac{n}{2(n-1)!}\\sum\\limits_{k=1}^n(-1)^k\\begin{pmatrix}n\\\\k\\end{pmatrix}(nx-k)^{n-1}{\\rm sgn}(nx-k).$$<\/p>\n

\"\"<\/a><\/p>\n

How about that.<\/p>\n

For what it’s worth, here is the Maxima<\/a> script to generate the Irwin-Hall plot:<\/p>\n

fI(x,n):=1\/2\/(n-1)!*sum((-1)^k*n!\/k!\/(n-k)!*(x-k)^(n-1)*signum(x-k),k,0,n);\r\nplot2d([fI(x,1),fI(x,2),fI(x,4),fI(x,8),fI(x,16)],[x,-2,18],[box,false],\r\n       [legend,\"n=1\",\"n=2\",\"n=4\",\"n=8\",\"n=16\"],[y,-0.1,1.1]);<\/pre>\n
And this one for the Bates plot:<\/p>\n
fB(x,n):=n\/2\/(n-1)!*sum((-1)^k*n!\/k!\/(n-k)!*(n*x-k)^(n-1)*signum(n*x-k),k,0,n);\r\nplot2d([fB(x,1),fB(x,2),fB(x,4),fB(x,8),fB(x,16)],[x,-0.1,1.1],[box,false],\r\n       [legend,\"n=1\",\"n=2\",\"n=4\",\"n=8\",\"n=16\"],[y,-0.1,5.9]);<\/pre>\n
Yes, I am still a little bit of a math geek at heart.<\/p>\n<\/fb:like>","protected":false},"excerpt":{"rendered":"
So here I am, reading about some trivial yet not-so-trivial probability distributions. Let’s start with the uniform distribution. Easy-peasy, isn’t it: a random number, between 0 and 1, with an equal probability assigned to any value within this range. So… what happens if I take two such random numbers and add them? Why, I get […]<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[30],"tags":[],"class_list":["post-9956","post","type-post","status-publish","format-standard","hentry","category-mathematics","category-30-id","post-seq-1","post-parity-odd","meta-position-corners","fix"],"_links":{"self":[{"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/posts\/9956","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=9956"}],"version-history":[{"count":11,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/posts\/9956\/revisions"}],"predecessor-version":[{"id":9969,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=\/wp\/v2\/posts\/9956\/revisions\/9969"}],"wp:attachment":[{"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=9956"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=9956"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/spinor.info\/weblog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=9956"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}