{"id":110964,"date":"2015-08-06T03:00:00","date_gmt":"2015-08-06T00:00:00","guid":{"rendered":"http:\/\/www.fyysika.ee\/?guid=d2790fcd5ad253f7804c6492614b595b"},"modified":"2015-08-06T03:00:00","modified_gmt":"2015-08-06T00:00:00","slug":"large-deviations-of-the-maximum-of-independent-and-identically-distributed-random-variables","status":"publish","type":"post","link":"https:\/\/www.fyysika.ee\/?p=110964","title":{"rendered":"Large deviations of the maximum of independent and identically distributed random variables"},"content":{"rendered":"<p>A pedagogical account of some aspects of extreme value statistics (EVS) is presented from the<br \/>\nsomewhat non-standard viewpoint of large deviation theory. We address the following problem: given a<br \/>\nset of N independent and identically distributed (i.i.d.) random variables ##IMG##<br \/>\n[http:\/\/ej.iop.org\/images\/0143-0807\/36\/5\/055037\/ejp518115ieqn1.gif] {${{X}_{1},ldots ,{X}_{N}}$}<br \/>\ndrawn from a parent probability density function (pdf) ##IMG##<br \/>\n[http:\/\/ej.iop.org\/images\/0143-0807\/36\/5\/055037\/ejp518115ieqn2.gif] {$p(x)$} , what is the<br \/>\nprobability that the maximum value of the set ##IMG##<br \/>\n[http:\/\/ej.iop.org\/images\/0143-0807\/36\/5\/055037\/ejp518115ieqn3.gif]<br \/>\n{${X}_{mathrm{max}}={mathrm{max}}_{i}{X}_{i}$} is \u2018atypically larger\u2019 than expected? The cases of<br \/>\nexponential and Gaussian distributed variables are worked out in detail, and the right rate function<br \/>\nfor a general pdf in the Gumbel basin of attraction is derived. The&#8230;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A pedagogical account of some aspects of extreme value statistics (EVS) is presented from the<br \/>\nsomewhat non-standard viewpoint of large deviation theory. We address the following problem: given a<br \/>\nset of N independent and identically distributed (i.i.d&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[178],"tags":[],"class_list":{"0":"post-110964","1":"post","2":"type-post","3":"status-publish","4":"format-standard","6":"category-rss-fuusikaharidus","7":"entry"},"_links":{"self":[{"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=\/wp\/v2\/posts\/110964","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=110964"}],"version-history":[{"count":0,"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=\/wp\/v2\/posts\/110964\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=110964"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=110964"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=110964"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}