{"id":85,"date":"2025-09-20T01:58:46","date_gmt":"2025-09-20T01:58:46","guid":{"rendered":"https:\/\/itisallmath.com\/probability-theory\/"},"modified":"2025-12-31T14:53:48","modified_gmt":"2025-12-31T14:53:48","slug":"probability-theory","status":"publish","type":"page","link":"https:\/\/itisallmath.com\/es\/mathematics\/probability-theory\/","title":{"rendered":"Probability Theory"},"content":{"rendered":"<div class=\"wp-block-uagb-container uagb-block-7e74bdf9 alignfull uagb-is-root-container\"><div class=\"uagb-container-inner-blocks-wrap\"><\/div><\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-89b27b49 alignfull uagb-is-root-container\"><div class=\"uagb-container-inner-blocks-wrap\">\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1600\" height=\"423\" src=\"https:\/\/itisallmath.com\/wp-content\/uploads\/2025\/09\/websitebannerprobabilitytheory-m5Kv56KEGJUw7QBv.webp\" alt=\"\" class=\"wp-image-50\" style=\"width:1200px;height:auto\" srcset=\"https:\/\/itisallmath.com\/wp-content\/uploads\/2025\/09\/websitebannerprobabilitytheory-m5Kv56KEGJUw7QBv.webp 1600w, https:\/\/itisallmath.com\/wp-content\/uploads\/2025\/09\/websitebannerprobabilitytheory-m5Kv56KEGJUw7QBv-300x79.webp 300w, https:\/\/itisallmath.com\/wp-content\/uploads\/2025\/09\/websitebannerprobabilitytheory-m5Kv56KEGJUw7QBv-1024x271.webp 1024w, https:\/\/itisallmath.com\/wp-content\/uploads\/2025\/09\/websitebannerprobabilitytheory-m5Kv56KEGJUw7QBv-768x203.webp 768w, https:\/\/itisallmath.com\/wp-content\/uploads\/2025\/09\/websitebannerprobabilitytheory-m5Kv56KEGJUw7QBv-1536x406.webp 1536w\" sizes=\"auto, (max-width: 1600px) 100vw, 1600px\" \/><\/figure>\n\n\n\n<div class=\"wp-block-uagb-advanced-heading uagb-block-b13d1ada\"><h1 class=\"uagb-heading-text\">What are you going to learn?<\/h1><\/div>\n\n\n\n<p>Extend and formalize knowledge of the theory of probability and random variables. Compute conditional probabilities directly and using Baye\u2019s theorem and check for independence of events. Perform probability calculations relating to probability distributions for discrete random variables. Perform probability calculations relating to probability density functions for continuous random variables. Compute Mathematical Expectation and variance. Apply various distributions to solve real life problems.<\/p>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-uagb-advanced-heading uagb-block-86673287\"><h2 class=\"uagb-heading-text\">Content<\/h2><\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-4ebe90d6 alignfull uagb-is-root-container\"><div class=\"uagb-container-inner-blocks-wrap\">\n<div class=\"wp-block-uagb-container uagb-block-f1cbfcf8\">\n<div class=\"wp-block-uagb-advanced-heading uagb-block-5a9d371a\"><h3 class=\"uagb-heading-text\"><strong>Chapter 1. Combinational Analysis<\/strong><\/h3><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Principles of Counting<\/li>\n\n\n\n<li>Permutations<\/li>\n\n\n\n<li>Combinations<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-8397072f\"><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-7bf0b0b6 alignfull uagb-is-root-container\"><div class=\"uagb-container-inner-blocks-wrap\">\n<div class=\"wp-block-uagb-container uagb-block-6d24f86d\">\n<div class=\"wp-block-uagb-advanced-heading uagb-block-e6b4bebc\"><h3 class=\"uagb-heading-text\"><strong>Chapter 2. Probability<\/strong><\/h3><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Introduction<\/li>\n\n\n\n<li>Sample Space and Events<\/li>\n\n\n\n<li>Axioms of Probability<\/li>\n\n\n\n<li>Probability as a Continuous Set Function<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-b1da5584\"><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-b6cd1ee5 alignfull uagb-is-root-container\"><div class=\"uagb-container-inner-blocks-wrap\">\n<div class=\"wp-block-uagb-container uagb-block-6beab58e\">\n<div class=\"wp-block-uagb-advanced-heading uagb-block-28646d74\"><h3 class=\"uagb-heading-text\"><strong><strong>Chapter 3. Conditional Probability and Independence<\/strong><\/strong><\/h3><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Introduction<\/li>\n\n\n\n<li>Conditional Probabilities<\/li>\n\n\n\n<li>Baye&#8217;s Formula<\/li>\n\n\n\n<li>Independent Events<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-09d3ff82\"><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-cc4ef321 alignfull uagb-is-root-container\"><div class=\"uagb-container-inner-blocks-wrap\">\n<div class=\"wp-block-uagb-container uagb-block-c3745215\">\n<div class=\"wp-block-uagb-advanced-heading uagb-block-013b92c4\"><h3 class=\"uagb-heading-text\"><strong>Chapter 4. Random Variables<\/strong><\/h3><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Introduction<\/li>\n\n\n\n<li>Random Variables<\/li>\n\n\n\n<li>Distribution Function<\/li>\n\n\n\n<li>Density Function<\/li>\n\n\n\n<li>Variance<\/li>\n\n\n\n<li>The Uniform Density Random Variable<\/li>\n\n\n\n<li>The Bernoulli and Binomial Random Variable<\/li>\n\n\n\n<li>The Poisson Random Variable<\/li>\n\n\n\n<li>Conditional Distribution\/Density Functions<\/li>\n\n\n\n<li>Operation with Random Variables<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-9a986f74\"><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-40bd1f8c alignfull uagb-is-root-container\"><div class=\"uagb-container-inner-blocks-wrap\">\n<div class=\"wp-block-uagb-container uagb-block-64d0a327\">\n<div class=\"wp-block-uagb-advanced-heading uagb-block-1da60346\"><h3 class=\"uagb-heading-text\"><strong><strong>Chapter 5. Operations on One Random Variable<\/strong><\/strong><\/h3><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Introduction<\/li>\n\n\n\n<li>Expectation<\/li>\n\n\n\n<li>Moments<\/li>\n\n\n\n<li>Moment Generating Functions<\/li>\n\n\n\n<li>Transformations of Random Variables<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-7373320e\"><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-e5b690df alignfull uagb-is-root-container\"><div class=\"uagb-container-inner-blocks-wrap\">\n<div class=\"wp-block-uagb-container uagb-block-7393ea03\">\n<div class=\"wp-block-uagb-advanced-heading uagb-block-1f4e2fda\"><h3 class=\"uagb-heading-text\"><strong><strong>Chapter 6. Multiple Random Variables<\/strong><\/strong><\/h3><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Introduction<\/li>\n\n\n\n<li>Joint Distribution Functions<\/li>\n\n\n\n<li>Conditional Distribution and Density<\/li>\n\n\n\n<li>Statistical Independence<\/li>\n\n\n\n<li>Sum of Independent Random Variables<\/li>\n\n\n\n<li>Conditional Distribution Functions<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-f59cc6af\"><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-9f6d7a94 alignfull uagb-is-root-container\"><div class=\"uagb-container-inner-blocks-wrap\">\n<div class=\"wp-block-uagb-container uagb-block-77ccb496\">\n<div class=\"wp-block-uagb-advanced-heading uagb-block-e4223a78\"><h3 class=\"uagb-heading-text\"><strong><strong>Chapter 7. Operation on Multiple Random Variables<\/strong><\/strong><\/h3><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Introduction<\/li>\n\n\n\n<li>Expected Value of a Function of Random Variables<\/li>\n\n\n\n<li>Joint Characteristic Functions<\/li>\n\n\n\n<li>Jointly Gaussian Random Variables<\/li>\n\n\n\n<li>Transformations of Multiple Random Variables<\/li>\n\n\n\n<li>Linear Transformations of Gaussian Random Variables<\/li>\n\n\n\n<li>Complex Random Variables<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-ed44a39d\"><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-5befdc29 alignfull uagb-is-root-container\"><div class=\"uagb-container-inner-blocks-wrap\">\n<div class=\"wp-block-uagb-container uagb-block-9194f6c5\">\n<div class=\"wp-block-uagb-advanced-heading uagb-block-9ea749eb\"><h3 class=\"uagb-heading-text\"><strong>Chapter 8. Limits Theorems<\/strong><\/h3><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Introduction<\/li>\n\n\n\n<li>Chebyshev&#8217;s Inequality<\/li>\n\n\n\n<li>The Central Limit Theorem<\/li>\n\n\n\n<li>The Strong Law of Large Numbers<\/li>\n\n\n\n<li>Other Inequalities<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-2ecc54da\"><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-c7f7a1d0 alignfull uagb-is-root-container\"><div class=\"uagb-container-inner-blocks-wrap\">\n<div class=\"wp-block-uagb-container uagb-block-72961fc3\">\n<div class=\"wp-block-uagb-advanced-heading uagb-block-76b05b60\"><h3 class=\"uagb-heading-text\"><strong>Chapter 9. Numerical Simulations<\/strong><\/h3><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Introduction<\/li>\n\n\n\n<li>General Techniques<\/li>\n\n\n\n<li>Simulating Discrete Distributions<\/li>\n\n\n\n<li>Variance Reduction Techniques<\/li>\n<\/ul>\n<\/div>\n\n\n\n<div class=\"wp-block-uagb-container uagb-block-f2131170\"><\/div>\n<\/div><\/div>\n\n\n\n<div class=\"wp-block-uagb-advanced-heading uagb-block-6b1c3c64\"><h2 class=\"uagb-heading-text\">Bibliography<\/h2><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Sheldom, R.&nbsp;<strong><em>A. First Course in Probability<\/em><\/strong>. 9th ed., Pearson, New York, 2010.<\/li>\n\n\n\n<li>Feller, William .&nbsp;<strong><em>An Introduction to Probability Theory and Its Applications, Vol. 1<\/em><\/strong>. 3d ed. John Wiley &amp; Sons, New York, 1968.<\/li>\n<\/ul>\n\n\n\n<div class=\"wp-block-uagb-advanced-heading uagb-block-12e3741d\"><h2 class=\"uagb-heading-text\">Bibliography<\/h2><\/div>\n\n\n\n<ul class=\"wp-block-list\">\n<li><\/li>\n<\/ul>","protected":false},"excerpt":{"rendered":"<p>What are you going to learn? Extend and formalize knowledge of the theory of probability and random variables. Compute conditional [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":340,"menu_order":12,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"_uag_custom_page_level_css":"","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"disabled","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"wf_page_folders":[47],"class_list":["post-85","page","type-page","status-publish","hentry"],"_hostinger_reach_plugin_has_subscription_block":false,"_hostinger_reach_plugin_is_elementor":false,"uagb_featured_image_src":{"full":false,"thumbnail":false,"medium":false,"medium_large":false,"large":false,"hd_qu_size2":false,"1536x1536":false,"2048x2048":false,"trp-custom-language-flag":false},"uagb_author_info":{"display_name":"carroyav02@gmail.com","author_link":"https:\/\/itisallmath.com\/es\/author\/carroyav02gmail-com\/"},"uagb_comment_info":0,"uagb_excerpt":"What are you going to learn? Extend and formalize knowledge of the theory of probability and random variables. Compute conditional [&hellip;]","_links":{"self":[{"href":"https:\/\/itisallmath.com\/es\/wp-json\/wp\/v2\/pages\/85","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/itisallmath.com\/es\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/itisallmath.com\/es\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/itisallmath.com\/es\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/itisallmath.com\/es\/wp-json\/wp\/v2\/comments?post=85"}],"version-history":[{"count":5,"href":"https:\/\/itisallmath.com\/es\/wp-json\/wp\/v2\/pages\/85\/revisions"}],"predecessor-version":[{"id":1549,"href":"https:\/\/itisallmath.com\/es\/wp-json\/wp\/v2\/pages\/85\/revisions\/1549"}],"up":[{"embeddable":true,"href":"https:\/\/itisallmath.com\/es\/wp-json\/wp\/v2\/pages\/340"}],"wp:attachment":[{"href":"https:\/\/itisallmath.com\/es\/wp-json\/wp\/v2\/media?parent=85"}],"wp:term":[{"taxonomy":"wf_page_folders","embeddable":true,"href":"https:\/\/itisallmath.com\/es\/wp-json\/wp\/v2\/wf_page_folders?post=85"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}