{"id":123,"date":"2023-12-27T06:06:24","date_gmt":"2023-12-27T06:06:24","guid":{"rendered":"https:\/\/eblog.armancomputer.com\/wp\/?p=123"},"modified":"2023-12-27T23:23:35","modified_gmt":"2023-12-27T23:23:35","slug":"normal-random-number-generation-statistical-distrubutions-and-r-commands","status":"publish","type":"post","link":"https:\/\/eblog.armancomputer.com\/wp\/2023\/12\/27\/normal-random-number-generation-statistical-distrubutions-and-r-commands\/","title":{"rendered":"Normal Random Number Generation | Statistical Distrubutions and R commands"},"content":{"rendered":"<p>The <a href=\"https:\/\/eblog.armancomputer.com\/wp\/2023\/12\/02\/ping\/\">statistical<\/a> approach to random numbers is very particular. In gambling and games of chance, random numbers have the main role. So, in a sequence of posts in <a href=\"http:\/\/armancomputer.com\">Armancomputer magazine<\/a>, I will discuss probability distribution and introduce some commands in R language, to generate <a href=\"https:\/\/eblog.armancomputer.com\/wp\/2023\/12\/19\/random-number-in-ms-excel-how-to-use\/\">random numbers<\/a> from the main statistical distributions such as Normal, Uniform, t-student, and chi-square.<\/p>\r\n<hr class=\"wp-block-separator has-alpha-channel-opacity\" \/>\r\n<h2>Random Number Generation and Statistical Distribution<\/h2>\r\n<p>In probability Theory, we compute probability based on the sample space and event set. As a rule, the probability of an event is its size (as a set) divided by the sample set size. However, this definition has some limitations.<\/p>\r\n<ul>\r\n<li>If sample space is not countable, how is probability computed.<\/li>\r\n<li>If the event is not countable, how is probability computed.<\/li>\r\n<li>&#8230;<\/li>\r\n<\/ul>\r\n<p style=\"text-align: left;\">And some many questions about real-world probability and random phenomena.<\/p>\r\n<ul>\r\n<li>is there any rule for computing probability in the real world?<\/li>\r\n<li>is there any pattern to see an event for the random phenomenon.<\/li>\r\n<li>&#8230;<\/li>\r\n<\/ul>\r\n<p>These problems, lead the scientists to a new approach to probability and its rules.<\/p>\r\n<h3>Statistical distributions and Random numbers<\/h3>\r\n<p>Many real random phenomena have a pattern of occurrence. This rule is named the Probability distribution. However using some translations, we convert a set function of probability (measure theory) to a real function to recognize these patterns or distributions. So, the set theory is converted to real function problems in probability and statistical theory.<\/p>\r\n<h3>Normal Distribution<\/h3>\r\n<p>If you come across a random phenomenon its outcome is real numbers, it may fit by Normal Distribution. The normal distribution is identified by the below plot.<\/p>\r\n<figure style=\"width: 459px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/7\/74\/Normal_Distribution_PDF.svg\"><img loading=\"lazy\" decoding=\"async\" class=\"\" title=\"Normal Distibution\" src=\"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/7\/74\/Normal_Distribution_PDF.svg\" alt=\"Normal Distibution\" width=\"459\" height=\"293\" \/><\/a><figcaption class=\"wp-caption-text\">Normal Distribution<\/figcaption><\/figure>\r\n<p>The plots show that there are two parameters for normal distribution.\u00a0 <a href=\"https:\/\/eblog.armancomputer.com\/wp\/2023\/12\/07\/arithmetic-mean-computation-and-benefits\/\">Mean<\/a> as shown by $\\mu$ and variance $\\sigma^2$. The <em><strong>rnorm<\/strong><\/em> command in R may generate random numbers from a normal distribution. As it is clear, <em><strong>rnorm<\/strong><\/em> has three parameters.<\/p>\r\n<pre class=\"lang:r decode:true\" title=\"Normal Random number generation command and its parameters\">rnorm(n,mean,sd)\r\nrnorm(100,0,1)<\/pre>\r\n<p>The first parameter (<em>n<\/em>) shows how many random numbers may be generated. The second and third parameters are normal distribution characteristics. Mean and Standard deviation (square root of variance). The R language code below shows how to generate normal random numbers and visualize their distribution.<\/p>\r\n<pre class=\"lang:r decode:true\" title=\"Generate Normal Random Numbers\">n = 10000\r\nmean = 0 \r\nsd = 1\r\nx=rnorm(n,mean,sd)<\/pre>\r\n<p>The probability distribution graph is plotted as follows:<\/p>\r\n<figure id=\"attachment_129\" aria-describedby=\"caption-attachment-129\" style=\"width: 503px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-129 \" src=\"https:\/\/eblog.armancomputer.com\/wp\/wp-content\/uploads\/2023\/12\/normal-density-plot-1024x742.png\" alt=\"normal density plot\" width=\"503\" height=\"365\" srcset=\"https:\/\/eblog.armancomputer.com\/wp\/wp-content\/uploads\/2023\/12\/normal-density-plot-1024x742.png 1024w, https:\/\/eblog.armancomputer.com\/wp\/wp-content\/uploads\/2023\/12\/normal-density-plot-300x217.png 300w, https:\/\/eblog.armancomputer.com\/wp\/wp-content\/uploads\/2023\/12\/normal-density-plot-768x556.png 768w, https:\/\/eblog.armancomputer.com\/wp\/wp-content\/uploads\/2023\/12\/normal-density-plot-600x435.png 600w, https:\/\/eblog.armancomputer.com\/wp\/wp-content\/uploads\/2023\/12\/normal-density-plot.png 1096w\" sizes=\"auto, (max-width: 503px) 100vw, 503px\" \/><figcaption id=\"caption-attachment-129\" class=\"wp-caption-text\">normal density plot<\/figcaption><\/figure>\r\n<p>PS: If using the ggplot2 library, your command may be as follows:<\/p>\r\n<pre class=\"lang:r decode:true\" title=\"Normal Random number Generation and its distribution\">n = 10000\r\nmean = 0 \r\nsd = 1\r\nx=rnorm(n,mean,sd)\r\nplot(density(x))\r\nlibrary(ggplot2)\r\nx = as.data.frame(x)\r\nggplot(x,aes(x = x)) +  geom_density(alpha = 0.5) \r\n<\/pre>\r\n<p>In the next post, I will describe how to compare two normal distributions through their plots. \u00a0 \u00a0<\/p><div class=\"pvc_clear\"><\/div><p id=\"pvc_stats_123\" class=\"pvc_stats all  \" data-element-id=\"123\" style=\"\"><i class=\"pvc-stats-icon small\" aria-hidden=\"true\"><svg aria-hidden=\"true\" focusable=\"false\" data-prefix=\"far\" data-icon=\"chart-bar\" role=\"img\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 512 512\" class=\"svg-inline--fa fa-chart-bar fa-w-16 fa-2x\"><path fill=\"currentColor\" d=\"M396.8 352h22.4c6.4 0 12.8-6.4 12.8-12.8V108.8c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v230.4c0 6.4 6.4 12.8 12.8 12.8zm-192 0h22.4c6.4 0 12.8-6.4 12.8-12.8V140.8c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v198.4c0 6.4 6.4 12.8 12.8 12.8zm96 0h22.4c6.4 0 12.8-6.4 12.8-12.8V204.8c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v134.4c0 6.4 6.4 12.8 12.8 12.8zM496 400H48V80c0-8.84-7.16-16-16-16H16C7.16 64 0 71.16 0 80v336c0 17.67 14.33 32 32 32h464c8.84 0 16-7.16 16-16v-16c0-8.84-7.16-16-16-16zm-387.2-48h22.4c6.4 0 12.8-6.4 12.8-12.8v-70.4c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v70.4c0 6.4 6.4 12.8 12.8 12.8z\" class=\"\"><\/path><\/svg><\/i> <img loading=\"lazy\" decoding=\"async\" width=\"16\" height=\"16\" alt=\"Loading\" src=\"https:\/\/eblog.armancomputer.com\/wp\/wp-content\/plugins\/page-views-count\/ajax-loader-2x.gif\" border=0 \/><\/p><div class=\"pvc_clear\"><\/div>","protected":false},"excerpt":{"rendered":"<p>The statistical approach to random numbers is very particular. In gambling and games of chance, random numbers have the main role. So, in a sequence of posts in Armancomputer magazine, I will discuss probability distribution and introduce some commands in R language, to generate random numbers from the main statistical distributions such as Normal, Uniform, [&hellip;]<\/p>\n<div class=\"pvc_clear\"><\/div>\n<p id=\"pvc_stats_123\" class=\"pvc_stats all  \" data-element-id=\"123\" style=\"\"><i class=\"pvc-stats-icon small\" aria-hidden=\"true\"><svg aria-hidden=\"true\" focusable=\"false\" data-prefix=\"far\" data-icon=\"chart-bar\" role=\"img\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 512 512\" class=\"svg-inline--fa fa-chart-bar fa-w-16 fa-2x\"><path fill=\"currentColor\" d=\"M396.8 352h22.4c6.4 0 12.8-6.4 12.8-12.8V108.8c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v230.4c0 6.4 6.4 12.8 12.8 12.8zm-192 0h22.4c6.4 0 12.8-6.4 12.8-12.8V140.8c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v198.4c0 6.4 6.4 12.8 12.8 12.8zm96 0h22.4c6.4 0 12.8-6.4 12.8-12.8V204.8c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v134.4c0 6.4 6.4 12.8 12.8 12.8zM496 400H48V80c0-8.84-7.16-16-16-16H16C7.16 64 0 71.16 0 80v336c0 17.67 14.33 32 32 32h464c8.84 0 16-7.16 16-16v-16c0-8.84-7.16-16-16-16zm-387.2-48h22.4c6.4 0 12.8-6.4 12.8-12.8v-70.4c0-6.4-6.4-12.8-12.8-12.8h-22.4c-6.4 0-12.8 6.4-12.8 12.8v70.4c0 6.4 6.4 12.8 12.8 12.8z\" class=\"\"><\/path><\/svg><\/i> <img loading=\"lazy\" decoding=\"async\" width=\"16\" height=\"16\" alt=\"Loading\" src=\"https:\/\/eblog.armancomputer.com\/wp\/wp-content\/plugins\/page-views-count\/ajax-loader-2x.gif\" border=0 \/><\/p>\n<div class=\"pvc_clear\"><\/div>\n","protected":false},"author":2,"featured_media":135,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3,55,4],"tags":[93,94,29,91,90,19,92,81,86,83,56,87,80,85,84,89,82,88],"class_list":["post-123","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-general","category-r-langauage","category-stat","tag-density-function","tag-density-plot","tag-descriptive-statistics","tag-ggplot-library","tag-ggplot2","tag-mean","tag-plot-by-ggplot","tag-probability-theory","tag-r-codes","tag-r-commands","tag-r-language","tag-r-tutorial","tag-random-numner","tag-rnorm","tag-rnorm-command","tag-standard-deviation","tag-statistics-theory","tag-variance"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v24.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Normal Random Number Generation | Statistical Distrubutions and R commands - Arman Computer Blog<\/title>\n<meta name=\"description\" content=\"Normal Random Number Generation and its command in R language will be discussed in this post. 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