{"id":26790,"date":"2021-06-13T13:00:12","date_gmt":"2021-06-13T13:00:12","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=26790"},"modified":"2024-05-22T03:26:08","modified_gmt":"2024-05-22T03:26:08","slug":"random-variables-and-probability-distributions","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/random-variables-and-probability-distributions\/","title":{"rendered":"Random Variables and Probability Distributions"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Random Variables and Probability Distributions<\/h1>\n<p><\/p>\n<p style=\"text-align:center;\"><strong>Summary<\/strong><br \/><em>This class provides an in-depth immersion into the concepts of random variables and probability distributions, fundamental pillars of probability theory and statistical analysis. The definition of a random variable as a number that depends on the outcome of a random experiment is introduced. The distribution function of a random variable is addressed, highlighting its importance as well as its essential properties. Finally, the relationship between random variables and probability distributions is analyzed, explaining that two variables can have the same distribution without being the same random variable.<\/em><\/p>\n<p><\/center><br \/>\n<\/p>\n<p style=\"text-align:center;\"><strong>LEARNING OBJECTIVES:<\/strong><br \/>\nBy the end of this class, the student will be able to:\n<\/p>\n<ol>\n<li><strong>Understand<\/strong> the concept of random variables: Students should be able to describe and explain what random variables are and how they are mathematically defined.<\/li>\n<li><strong>Understand<\/strong> the concept of probability distributions: Students should be able to explain what probability distributions are and how they are represented.<\/li>\n<li><strong>Describe<\/strong> the properties of probability distributions: Students should be able to recognize and explain the key properties of probability distributions.<\/li>\n<li><strong>Analyze<\/strong> the relationship between random variables and probability distributions: Students should be able to discuss how random variables and probability distributions are interrelated and how two variables can have the same distribution without being the same random variable.<\/li>\n<li><strong>Demonstrate<\/strong> and apply the properties of probability distributions in practical situations: Students should be able to mathematically demonstrate the properties of probability distributions and apply these properties in real-world situations.<\/li>\n<li><strong>Understand<\/strong> the concept of distribution functions: Students should be able to describe what a distribution function is and how it is used to describe a random variable.<\/li>\n<\/ol>\n<p><center><br \/>\n<strong><u>CONTENTS INDEX<\/u>:<\/strong><br \/>\n<a href=\"#1\"><strong>What are random variables?<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>What are probability distributions?<\/strong><\/a><br \/>\n<a href=\"#3\">Properties of probability distributions<\/a><br \/>\n<a href=\"#4\">Relationship between random variables and probability distributions<\/a><br \/>\n<\/center><br \/>\n<center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/gIKn9t1hnrw\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center>\n<\/div>\n<p style=\"text-align: justify; color: #000000;\">One of the key concepts of probability theory and statistical analysis is that of random variables and probability distributions. Although the theory we have developed so far is in a sense \u00abcomplete,\u00bb the truth is that in its current state it is quite rudimentary; random variables and probability distributions are, so to speak, concepts that allow us to \u00aboil our ability to work with probabilities and perform statistical analysis.\u00bb<\/p>\n<p><a name=\"1\"><\/a><\/br><\/br><\/p>\n<h2>What are random variables?<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><strong><span style=\"color: #ff0000;\">To familiarize ourselves with the concept of a random variable<\/span><\/strong>, it is useful to start with an intuitive approach: a random variable can be interpreted as \u00aba number that depends on the outcome of a random experiment.\u00bb However, for a more precise understanding, it is essential to also explore its formal definition. Let&#8217;s see this definition:<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0ffe0;\">\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #000080;\"><strong>Definition:<\/strong><\/span> A random variable over a set <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{X}<\/span><\/span> is a function <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">f:\\Omega \\longmapsto \\mathcal{X}<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">The most common case is when <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\mathcal{X}= \\mathbb{R},<\/span><\/span> and unless otherwise specified, this is what we will assume from now on; that is, we will work with real-valued random variables. Generally, random variables are denoted by uppercase letters, such as <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X,Y,Z, \\cdots,<\/span><\/span> while constants are denoted by lowercase letters. For simplicity, we will refer to random variables simply as \u00abvariables.\u00bb<\/p>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0e0ff;\">\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #000000;\"><strong>Example:<\/strong><\/span> Suppose a 6-sided die is rolled twice. Then we have:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega_{2d6} = \\{(\\omega_1, \\omega_2)\\;|\\; \\omega_1,\\omega_2 \\in \\{1,2,3,4,5,6\\}\\}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">From this, we can define the following random variables:<\/p>\n<ul style=\"text-align: justify; color: #000000;\">\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X=<\/span><\/span> \u00abThe number of times one appears\u00bb<\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Y=<\/span><\/span> \u00abThe sum of the results obtained\u00bb <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">=\\omega_1 + \\omega_2<\/span><\/span><\/li>\n<li><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Z=<\/span><\/span> \u00abThe result of the second roll\u00bb <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">= \\omega_2<\/span><\/span><\/li>\n<\/ul>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"2\"><\/a><\/br><\/br><\/p>\n<h2>What are probability distributions?<\/h2>\n<table>\n<tbody>\n<tr>\n<td style=\"background-color: #e0ffe0;\">\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #000080;\"><strong>Definition: <\/strong><\/span><strong><span style=\"color: #ff0000;\">A distribution function (or \u00abDF\u00bb)<\/span><\/strong> of a random variable <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> is a function <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X: \\mathbb{R} \\longmapsto \\mathbb{R}<\/span><\/span> defined by the relation <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X(x) = P(\\{\\omega \\;|\\; X(\\omega)\\leq x\\}),<\/span><\/span> or more briefly: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X\\leq x).<\/span><\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify; color: #000000;\">Generally, what interests us about a random variable is not so much its explicit expression in a sample space <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span><\/span>, but its distribution function. The subscript <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> in <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F_X<\/span><\/span> can be omitted if the context is clear and there is no ambiguity. It is common to use the notation <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X\\sim F<\/span><\/span> to indicate that the random variable <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> has a distribution function <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F.<\/span><\/span><\/p>\n<p><a name=\"3\"><\/a><\/br><\/br><\/p>\n<h3>Properties of probability distributions<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=gIKn9t1hnrw&amp;t=806s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">If <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> is a probability distribution<\/span><\/strong><\/a> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a,b<\/span><\/span> are any real numbers, then the following properties will hold:<\/p>\n<p style=\"text-align: justify; color: #000000;\">(a) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\lt b \\longrightarrow [P(a\\lt X \\leq b) = F(b) - F(a)]<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">(b) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\lt b \\longrightarrow F(a) \\leq F(b),<\/span><\/span> i.e., \u00abF is increasing.\u00bb<\/p>\n<p style=\"text-align: justify; color: #000000;\">(c) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to +\\infty} F(x) = 1<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\lim_{x\\to -\\infty} F(x) = 0<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">(d) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P(X=x)=\\lim_{t\\to x^+}F(t) - \\lim_{t\\to x^-}F(t)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">(e) <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle F(x)=\\lim_{t\\to x^+}F(t)<\/span><\/span><\/p>\n<table style=\"text-align: justify; color: #000000;\">\n<tbody>\n<tr>\n<td style=\"background-color: #ffe0e0;\"><span class=\"collapseomatic \" id=\"id69e4336657719\"  tabindex=\"0\" title=\"PROOF\"    >PROOF<\/span><div id=\"target-id69e4336657719\" class=\"collapseomatic_content \">\n<p style=\"text-align: justify; color: #000000;\"><strong>(a)<\/strong> Let <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B<\/span><\/span> be the events <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{X\\leq a\\}<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\{X\\leq b\\}<\/span><\/span> respectively, with <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a\\lt b.<\/span><\/span> If all this happens, then we have <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A\\subseteq B<\/span><\/span> and therefore:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\color{blue}{P(a\\lt X\\leq b)} = P(B\\setminus A) = P(B) - P(B\\cap A) = P(B)-P(A) =\\color{blue}{F(b) - F(a)}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\"><strong>(b)<\/strong> From part (a) we have: Since <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(B\\setminus A)\\geq 0,<\/span><\/span> we have:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(b) - F(a) \\geq 0<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">which is the same as saying:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F(a) \\leq F(b)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\"><strong>(c)<\/strong> Here we will use the fact that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">F<\/span><\/span> is monotone increasing (proven in (b)) and bounded with a maximum value equal to \u00ab1\u00bb (because the distribution is defined in terms of probability). This alone is enough to say that:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{x\\to +\\infty} F(x) = 1<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">A complementary approach allows us to make the following calculations with the same result.<\/p>\n<p style=\"text-align: justify; color: #000000;\">Let&#8217;s define the set <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A_n=\\{\\omega\\;|\\;X(\\omega)\\leq n\\}.<\/span><\/span> From this, it is easy to verify that for any <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n<\/span><\/span> we have <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">A_{n}\\subseteq A_{n+1},<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\bigcup_{n\\lt +\\infty} A_n = \\Omega<\/span><\/span> and therefore, using the property of <a href=\"http:\/\/toposuranos.com\/material\/en\/useful-theorems-for-probability-calculation\/\" rel=\"noopener\" target=\"_blank\">continuity<\/a>, we have:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle 1=P(\\Omega) = P\\left( \\bigcup_{n\\lt +\\infty} A_n \\right) = \\lim_{n\\to +\\infty} P(A_n) = \\lim_{n\\to +\\infty} P(\\{\\omega\\;|\\;X(\\omega)\\leq n\\}) = \\lim_{n\\to +\\infty} P(X\\leq n)=\\lim_{n\\to +\\infty}F(n)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">That is:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\color{blue}{\\lim_{x\\to +\\infty} F(x) = 1}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Conversely, for the limit as <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x\\to -\\infty<\/span><\/span>, we have the following:<\/p>\n<p style=\"text-align: justify; color: #000000;\">First, let&#8217;s define the set <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">B_n=\\{\\omega\\;|\\;-n\\lt X(\\omega)\\}.<\/span><\/span> From this, it is verified that:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{n \\to -\\infty}F(n) = \\lim_{n\\to -\\infty} P(X\\leq n) = \\lim_{n\\to \\infty} P(X\\leq -n)= 1 - \\lim_{n\\to \\infty} P(-n \\lt X) = 1 - \\lim_{n\\to \\infty}P(B_n)) = 1 - P(\\Omega) = 1-1=0<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\"><strong>(d)<\/strong> The reasoning is similar to part (c). Let&#8217;s start by defining the set:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle C_n = \\left\\{x - \\frac{1}{n} \\leq X \\leq x + \\frac{1}{n}\\right\\}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">And from this, we have:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\"> C_{n+1}\\subseteq C_n<\/span><\/span><\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\bigcap_{n\\gt 0} C_n = \\{X=x\\}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Therefore, using <a href=\"http:\/\/toposuranos.com\/material\/en\/useful-theorems-for-probability-calculation\/\" rel=\"noopener\" target=\"_blank\">a result from the continuity property<\/a>, we have:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P(X=x)=P\\left(\\bigcap_{n\\gt 0} C_n \\right) = \\lim_{n\\to \\infty} P(C_n) = \\lim_{x+1\/n \\to x^+}F\\left(x+1\/n\\right) - \\lim_{x-1\/n \\to x^-}F\\left(x-1\/n\\right)= \\lim_{t \\to x^+}F\\left(t\\right) - \\lim_{t \\to x^-}F\\left(t\\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\"><strong>(e)<\/strong> This last case is obtained from the previous result. In fact, since we have already proved:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P(X=x)= \\lim_{t \\to x^+}F\\left(t\\right) - \\lim_{t \\to x^-}F\\left(t\\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">We can write:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\lim_{t \\to x^+}F\\left(t\\right) = P(X=x) + \\lim_{t \\to x^-}F\\left(t\\right) = P(X=x) + \\lim_{t\\to x^-}P(X\\leq t)= P(X\\leq x) = F(x)<\/span><\/span><\/p>\n<\/div><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><a name=\"4\"><\/a><\/br><\/br><\/p>\n<h3>Relationship between random variables and probability distributions<\/h3>\n<p style=\"text-align: justify; color: #000000;\">Two variables <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Y<\/span><\/span> are said to have the same probability distribution if <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(\\forall A\\subseteq \\mathbb{R})(P(X\\in A) = P(Y\\in A)).<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Two variables <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Y<\/span><\/span> defined on the same sample space <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span><\/span> can have the same distribution but not necessarily be the same random variable. For example, if we consider the experiment of flipping a fair two-sided coin and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X=1<\/span><\/span> corresponds to heads and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">X=0<\/span><\/span> corresponds to tails, we can define the random variable <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Y=1-X<\/span><\/span> and have <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X=1) = P(Y=1)=0.5,<\/span><\/span> and both have the same distribution, but if we calculate the probability that both have the same value, we have <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(X=Y)=0<\/span><\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Random Variables and Probability Distributions SummaryThis class provides an in-depth immersion into the concepts of random variables and probability distributions, fundamental pillars of probability theory and statistical analysis. The definition of a random variable as a number that depends on the outcome of a random experiment is introduced. The distribution function of a random variable [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":26787,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":1,"footnotes":""},"categories":[567,670],"tags":[],"class_list":["post-26790","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics","category-probabilities-and-statistics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Random Variables and Probability Distributions - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Discover the essential concepts of random variables and probability distributions in this detailed class.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/toposuranos.com\/material\/en\/random-variables-and-probability-distributions\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Random Variables and Probability Distributions\" \/>\n<meta property=\"og:description\" content=\"Discover the essential concepts of random variables and probability distributions in this detailed class.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/en\/random-variables-and-probability-distributions\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-06-13T13:00:12+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-05-22T03:26:08+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/Variables-Aleatorias-y-Distribuciones-de-Probabilidades.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Random Variables and Probability Distributions\" \/>\n<meta name=\"twitter:description\" content=\"Discover the essential concepts of random variables and probability distributions in this detailed class.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/Variables-Aleatorias-y-Distribuciones-de-Probabilidades.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"6 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\/\/toposuranos.com\/material\/en\/random-variables-and-probability-distributions\/#article\",\"isPartOf\":{\"@id\":\"http:\/\/toposuranos.com\/material\/en\/random-variables-and-probability-distributions\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Random Variables and Probability Distributions\",\"datePublished\":\"2021-06-13T13:00:12+00:00\",\"dateModified\":\"2024-05-22T03:26:08+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\/\/toposuranos.com\/material\/en\/random-variables-and-probability-distributions\/\"},\"wordCount\":1550,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\/\/toposuranos.com\/material\/#organization\"},\"image\":{\"@id\":\"http:\/\/toposuranos.com\/material\/en\/random-variables-and-probability-distributions\/#primaryimage\"},\"thumbnailUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/06\/Variables-Aleatorias-y-Distribuciones-de-Probabilidades.jpg\",\"articleSection\":[\"Mathematics\",\"Probabilities and Statistics\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\/\/toposuranos.com\/material\/en\/random-variables-and-probability-distributions\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\/\/toposuranos.com\/material\/en\/random-variables-and-probability-distributions\/\",\"url\":\"http:\/\/toposuranos.com\/material\/en\/random-variables-and-probability-distributions\/\",\"name\":\"Random Variables and Probability Distributions - 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