{"id":1014,"date":"2010-12-08T20:30:11","date_gmt":"2010-12-08T18:30:11","guid":{"rendered":"http:\/\/djalil.chafai.net\/blog\/?p=1014"},"modified":"2010-12-11T09:30:37","modified_gmt":"2010-12-11T07:30:37","slug":"double-exponential","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2010\/12\/08\/double-exponential\/","title":{"rendered":"Double exponential"},"content":{"rendered":"<p style=\"text-align: justify;\">We have already mentioned in a <a title=\"Exponential mixtures of exponentials are Pareto\" href=\"\/blog\/2010\/04\/30\/exponential-mixtures-of-exponentials-are-pareto\/\">previous post<\/a> an amusing property of the exponential distribution. Here is another one: if \\( X \\) and \\( Y \\) are two independent exponential random variables with mean \\( 1\/\\lambda \\) and \\( 1\/\\mu \\) respectively then \\( X-Y \\)\u00a0 follows the double exponential distribution on \\( \\mathbb{R} \\) with\u00a0 density<\/p>\n<p style=\"text-align: justify;\">\\[ x\\in\\mathbb{R}\\mapsto \\frac{\\lambda\\mu}{\\lambda+\\mu}\\left(e^{\\mu x}\\mathbf{1}_{\\mathbb{R}_-}(x)+e^{-\\lambda x}\\mathbf{1}_{\\mathbb{R}_+}(x)\\right). \\]<\/p>\n<p style=\"text-align: justify;\">In other words, we have the mixture \\[\\mathcal{L}(X-Y)=\\mathcal{L}(X)*\\mathcal{L}(-Y)=\\frac{\\mu}{\\lambda+\\mu}\\mathcal{L}(X)+\\frac{\\lambda}{\\lambda+\\mu}\\mathcal{L}(-Y).\\]<\/p>\n<p style=\"text-align: justify;\">In particular, when \\( \\lambda=\\mu \\) we get the symmetric double exponential (Laplace distribution) with density \\(x\\mapsto \\frac{\\lambda}{2} e^{-\\lambda|x|}\\).\u00a0 Another way to state the property is to say that the double exponential is the image of the product distribution \\( \\mathcal{E}(\\lambda)\\otimes\\mathcal{E}(\\mu) \\) by the linear map \\((x,y)\\mapsto x-y\\). Note that the density of \\(X+Y\\) is \\[x\\mapsto \\lambda\\mu\\frac{e^{-\\mu x}-e^{-\\lambda x}}{\\lambda-\\mu}\\mathbf{1}_{\\mathbb{R}_+}(x)\\]<\/p>\n<p style=\"text-align: justify;\">(when \\(\\lambda=\\mu\\) we recover by continuity the Gamma density \\(x\\mapsto \\lambda^2x e^{-\\lambda x} \\mathbf{1}_{\\mathbb{R}_+}(x)\\)).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>We have already mentioned in a previous post an amusing property of the exponential distribution. Here is another one: if \\( X \\) and \\(&#8230;<\/p>\n<div class=\"more-link-wrapper\"><a class=\"more-link\" href=\"https:\/\/djalil.chafai.net\/blog\/2010\/12\/08\/double-exponential\/\">Continue reading<span class=\"screen-reader-text\">Double exponential<\/span><\/a><\/div>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":52},"categories":[1],"tags":[],"_links":{"self":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/1014"}],"collection":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/comments?post=1014"}],"version-history":[{"count":21,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/1014\/revisions"}],"predecessor-version":[{"id":1064,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/posts\/1014\/revisions\/1064"}],"wp:attachment":[{"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/media?parent=1014"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/categories?post=1014"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/djalil.chafai.net\/blog\/wp-json\/wp\/v2\/tags?post=1014"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}