{"id":61,"date":"2024-11-04T19:02:49","date_gmt":"2024-11-04T11:02:49","guid":{"rendered":"https:\/\/www.toothlessos.xyz\/?p=61"},"modified":"2024-11-05T10:34:14","modified_gmt":"2024-11-05T02:34:14","slug":"recap-of-special-probability-distributions1","status":"publish","type":"post","link":"https:\/\/www.toothlessos.xyz\/index.php\/2024\/11\/04\/recap-of-special-probability-distributions1\/","title":{"rendered":"Recap of special probability distributions(1)"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">This and the following log will be a two part recap of special probability distributions. We will go over the formulation of the distributions and some typical modeling cases together.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Below is a mind map of the special probability distributions from Oliver Ibe&#8217;s <em>Fundamentals of Applied Probability and Random Processes<\/em>.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"767\" src=\"http:\/\/38.246.252.17:8080\/wp-content\/uploads\/2024\/11\/image-1024x767.png\" alt=\"\" class=\"wp-image-62\" srcset=\"https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/image-1024x767.png 1024w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/image-300x225.png 300w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/image-768x575.png 768w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/image.png 1416w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">For part 1, we will be mainly focusing on <strong>discrete random variables<\/strong>.<\/p>\n\n\n<div class=\"wp-block-aioseo-table-of-contents\"><ul><li><a class=\"aioseo-toc-item\" href=\"#aioseo-bernoulli-distribution\">Bernoulli distribution<\/a><\/li><li><a class=\"aioseo-toc-item\" href=\"#aioseo-binomial-distribution\">Binomial distribution<\/a><\/li><li><a class=\"aioseo-toc-item\" href=\"#aioseo-geometric-distribution\">Geometric distribution<\/a><ul><li><a class=\"aioseo-toc-item\" href=\"#aioseo-modified-geometric-distribution\">Modified geometric distribution<\/a><\/li><li><a class=\"aioseo-toc-item\" href=\"#aioseo-the-derivative-trick-for-sums\">The derivative trick for sums<\/a><\/li><li><a class=\"aioseo-toc-item\" href=\"#aioseo-the-memoryless-property\">The memoryless property<\/a><\/li><\/ul><\/li><li><a class=\"aioseo-toc-item\" href=\"#aioseo-pascal-k-distribution\">Pascal-k Distribution<\/a><\/li><li><a class=\"aioseo-toc-item\" href=\"#aioseo-poisson-distribution\">Poisson distribution<\/a><\/li><li><a class=\"aioseo-toc-item\" href=\"#aioseo-binomial-to-poisson-to-exponential\">Binomial to Poisson to Exponential<\/a><\/li><\/ul><\/div>\n\n\n<h2 class=\"wp-block-heading\" id=\"aioseo-bernoulli-distribution\">Bernoulli distribution<\/h2>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"291\" src=\"http:\/\/38.246.252.17:8080\/wp-content\/uploads\/2024\/11\/d121a649a94e61f9e002d93b02c1501-e1730710486497-1024x291.jpg\" alt=\"\" class=\"wp-image-66\" srcset=\"https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/d121a649a94e61f9e002d93b02c1501-e1730710486497-1024x291.jpg 1024w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/d121a649a94e61f9e002d93b02c1501-e1730710486497-300x85.jpg 300w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/d121a649a94e61f9e002d93b02c1501-e1730710486497-768x218.jpg 768w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/d121a649a94e61f9e002d93b02c1501-e1730710486497-1536x436.jpg 1536w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/d121a649a94e61f9e002d93b02c1501-e1730710486497.jpg 1963w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"aioseo-binomial-distribution\">Binomial distribution<\/h2>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"283\" src=\"http:\/\/38.246.252.17:8080\/wp-content\/uploads\/2024\/11\/d121a649a94e61f9e002d93b02c1501-1-e1730710537872-1024x283.jpg\" alt=\"\" class=\"wp-image-67\" srcset=\"https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/d121a649a94e61f9e002d93b02c1501-1-e1730710537872-1024x283.jpg 1024w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/d121a649a94e61f9e002d93b02c1501-1-e1730710537872-300x83.jpg 300w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/d121a649a94e61f9e002d93b02c1501-1-e1730710537872-768x212.jpg 768w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/d121a649a94e61f9e002d93b02c1501-1-e1730710537872-1536x425.jpg 1536w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/d121a649a94e61f9e002d93b02c1501-1-e1730710537872.jpg 1863w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"aioseo-geometric-distribution\">Geometric distribution<\/h2>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"467\" src=\"http:\/\/38.246.252.17:8080\/wp-content\/uploads\/2024\/11\/4a69b75e208b8cf89dbc11aeda378e6-e1730710742764-1024x467.jpg\" alt=\"\" class=\"wp-image-68\" srcset=\"https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/4a69b75e208b8cf89dbc11aeda378e6-e1730710742764-1024x467.jpg 1024w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/4a69b75e208b8cf89dbc11aeda378e6-e1730710742764-300x137.jpg 300w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/4a69b75e208b8cf89dbc11aeda378e6-e1730710742764-768x350.jpg 768w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/4a69b75e208b8cf89dbc11aeda378e6-e1730710742764.jpg 1366w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"aioseo-modified-geometric-distribution\">Modified geometric distribution<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Finding counts until first failure instead; Just exchange p &amp; (1-p) in the formulation.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"aioseo-the-derivative-trick-for-sums\">The derivative trick for sums<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">When computing the expectation and variance of the geometric distribution, we can construct a derivative form. This form allows us to transform the computation from &#8216;sum of derivatives&#8217; to &#8216;derivatives of sum&#8217;. In this case, it helps us the address the infinite sum and simplifies the calculation greatly. Below is the example:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Computation of the first moment E[X] =&gt;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"574\" src=\"http:\/\/38.246.252.17:8080\/wp-content\/uploads\/2024\/11\/36bc81e9f65a664c4a3546641c0da7c-e1730711617135-1024x574.jpg\" alt=\"\" class=\"wp-image-70\" srcset=\"https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/36bc81e9f65a664c4a3546641c0da7c-e1730711617135-1024x574.jpg 1024w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/36bc81e9f65a664c4a3546641c0da7c-e1730711617135-300x168.jpg 300w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/36bc81e9f65a664c4a3546641c0da7c-e1730711617135-768x430.jpg 768w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/36bc81e9f65a664c4a3546641c0da7c-e1730711617135-1536x861.jpg 1536w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/36bc81e9f65a664c4a3546641c0da7c-e1730711617135.jpg 1893w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Computation of the second moment E[X<sup>2<\/sup>] =&gt;<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"500\" src=\"http:\/\/38.246.252.17:8080\/wp-content\/uploads\/2024\/11\/2006387650eafc20eb7494961510112-e1730711631330-1024x500.jpg\" alt=\"\" class=\"wp-image-71\" srcset=\"https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/2006387650eafc20eb7494961510112-e1730711631330-1024x500.jpg 1024w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/2006387650eafc20eb7494961510112-e1730711631330-300x146.jpg 300w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/2006387650eafc20eb7494961510112-e1730711631330-768x375.jpg 768w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/2006387650eafc20eb7494961510112-e1730711631330-1536x750.jpg 1536w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/2006387650eafc20eb7494961510112-e1730711631330.jpg 1653w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<h3 class=\"wp-block-heading\" id=\"aioseo-the-memoryless-property\">The memoryless property<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">If we know that there were no success in the previous n attempts, the probability for the next k attempt from n to be the first successful one is the same as starting from next k attempt from zero. (Note that this the memoryless property <strong>may be reflect cases in real life correctly<\/strong>, for instance, a series of earthquakes is highly likely to be related):<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"213\" src=\"http:\/\/38.246.252.17:8080\/wp-content\/uploads\/2024\/11\/e394ad365f5ee8ddf2959d970fa5fff-e1730713011492-1024x213.jpg\" alt=\"\" class=\"wp-image-77\" srcset=\"https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/e394ad365f5ee8ddf2959d970fa5fff-e1730713011492-1024x213.jpg 1024w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/e394ad365f5ee8ddf2959d970fa5fff-e1730713011492-300x63.jpg 300w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/e394ad365f5ee8ddf2959d970fa5fff-e1730713011492-768x160.jpg 768w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/e394ad365f5ee8ddf2959d970fa5fff-e1730713011492.jpg 1425w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"391\" src=\"http:\/\/38.246.252.17:8080\/wp-content\/uploads\/2024\/11\/62383b1089ff7d9267d7f6675b395e0-e1730712982486-1024x391.jpg\" alt=\"\" class=\"wp-image-76\" srcset=\"https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/62383b1089ff7d9267d7f6675b395e0-e1730712982486-1024x391.jpg 1024w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/62383b1089ff7d9267d7f6675b395e0-e1730712982486-300x115.jpg 300w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/62383b1089ff7d9267d7f6675b395e0-e1730712982486-768x293.jpg 768w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/62383b1089ff7d9267d7f6675b395e0-e1730712982486-1536x586.jpg 1536w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/62383b1089ff7d9267d7f6675b395e0-e1730712982486.jpg 1789w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Here&#8217;s a graph illustrating the memoryless of exponential distribution (<a href=\"https:\/\/zhuanlan.zhihu.com\/p\/560955503\">\u6307\u6570\u5206\u5e03\u300cExponential Distribution\u300d &#8211; \u77e5\u4e4e<\/a>), which offers a geometric approach to memoryless property &#8211; considering the <strong>similarity<\/strong> of shapes.<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"700\" height=\"311\" src=\"http:\/\/38.246.252.17:8080\/wp-content\/uploads\/2024\/11\/image.jpeg\" alt=\"\" class=\"wp-image-89\" srcset=\"https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/image.jpeg 700w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/image-300x133.jpeg 300w\" sizes=\"auto, (max-width: 700px) 100vw, 700px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"aioseo-pascal-k-distribution\">Pascal-k Distribution<\/h2>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"938\" src=\"http:\/\/38.246.252.17:8080\/wp-content\/uploads\/2024\/11\/7ed4a708c3b03497218fa0f5e1f040a-e1730714699400-1024x938.jpg\" alt=\"\" class=\"wp-image-83\" srcset=\"https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/7ed4a708c3b03497218fa0f5e1f040a-e1730714699400-1024x938.jpg 1024w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/7ed4a708c3b03497218fa0f5e1f040a-e1730714699400-300x275.jpg 300w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/7ed4a708c3b03497218fa0f5e1f040a-e1730714699400-768x703.jpg 768w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/7ed4a708c3b03497218fa0f5e1f040a-e1730714699400.jpg 1166w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"aioseo-poisson-distribution\">Poisson distribution<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">For modeling the occurrence (k) of an event within a period of time. The parameter lambda  <\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"232\" src=\"http:\/\/38.246.252.17:8080\/wp-content\/uploads\/2024\/11\/image-1-1024x232.png\" alt=\"\" class=\"wp-image-84\" srcset=\"https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/image-1-1024x232.png 1024w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/image-1-300x68.png 300w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/image-1-768x174.png 768w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/image-1.png 1132w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>The value of both expectation and variance is lambda.<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A trick of memorization: The Taylor series of e<sup>x<\/sup> (Since you are summing for to check the normalization condition)<\/p>\n\n\n\n<h2 class=\"wp-block-heading\" id=\"aioseo-binomial-to-poisson-to-exponential\">Binomial to Poisson to Exponential<\/h2>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"835\" height=\"1024\" src=\"http:\/\/38.246.252.17:8080\/wp-content\/uploads\/2024\/11\/814a268da33a5047ce81c92605db953-e1730718012688-835x1024.jpg\" alt=\"\" class=\"wp-image-86\" srcset=\"https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/814a268da33a5047ce81c92605db953-e1730718012688-835x1024.jpg 835w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/814a268da33a5047ce81c92605db953-e1730718012688-245x300.jpg 245w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/814a268da33a5047ce81c92605db953-e1730718012688-768x942.jpg 768w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/814a268da33a5047ce81c92605db953-e1730718012688.jpg 1191w\" sizes=\"auto, (max-width: 835px) 100vw, 835px\" \/><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"934\" src=\"http:\/\/38.246.252.17:8080\/wp-content\/uploads\/2024\/11\/f0829d847c277b8bf9ac1a5c70fb5bf-e1730717990822-1024x934.jpg\" alt=\"\" class=\"wp-image-87\" srcset=\"https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/f0829d847c277b8bf9ac1a5c70fb5bf-e1730717990822-1024x934.jpg 1024w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/f0829d847c277b8bf9ac1a5c70fb5bf-e1730717990822-300x274.jpg 300w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/f0829d847c277b8bf9ac1a5c70fb5bf-e1730717990822-768x701.jpg 768w, https:\/\/www.toothlessos.xyz\/wp-content\/uploads\/2024\/11\/f0829d847c277b8bf9ac1a5c70fb5bf-e1730717990822.jpg 1276w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>This and the following log will be a two part recap of  [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[10,11],"class_list":["post-61","post","type-post","status-publish","format-standard","hentry","category-math","tag-probability-and-statistics","tag-probability-distributions"],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 4.9.8 - 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