{"id":28487,"date":"2022-08-30T13:33:38","date_gmt":"2022-08-30T13:33:38","guid":{"rendered":"https:\/\/analystprep.com\/study-notes\/?p=28487"},"modified":"2022-09-14T11:42:00","modified_gmt":"2022-09-14T11:42:00","slug":"explain-and-perform-calculations-concerning-joint-probability-functions-and-cumulative-distribution-functions-for-discrete-random-variables-only","status":"publish","type":"post","link":"https:\/\/analystprep.com\/study-notes\/actuarial-exams\/soa\/p-probability\/multivariate-random-variables\/explain-and-perform-calculations-concerning-joint-probability-functions-and-cumulative-distribution-functions-for-discrete-random-variables-only\/","title":{"rendered":"Explain and perform calculations concerning joint probability functions and cumulative distribution functions for discrete random variables only"},"content":{"rendered":"

Joint Discrete Probability Distributions<\/h2>\n

We are often interested in experiments that involve the intersection of two or more events.\u00a0<\/span><\/p>\n

For example:<\/span><\/p>\n

    \n
  1. An experimenter tossing a fair die is interested in the intersection of getting, say, a 5 and a 6.<\/span><\/li>\n
  2. The intersection of events of height and weight measures and so on.<\/span><\/li>\n<\/ol>\n

    Definition<\/strong><\/p>\n

    Let \\(X\\) and \\(Y\\) be two discrete random variables defined on a two-dimensional discrete space, \\(S\\).\u00a0<\/span><\/p>\n

    The <\/span>joint probability mass function <\/b>of \\(X\\) and \\(Y\\)<\/span>\u00a0is defined as<\/span><\/p>\n

    $$f(x, y) = P(X = x, Y = y)$$<\/span><\/p>\n

    Properties of a Joint Discrete Distribution<\/b><\/p>\n

      \n
    1. \\(0 \\leq f(x, y) \\leq 1\\); It means that the probability of any possible outcome must not be less than 0 and greater than 1.<\/li>\n
    2. \u00a0\\(\\sum_{\\mathrm{x}} \\sum_{\\mathrm{y}} \\mathrm{f}(\\mathrm{x}, \\mathrm{y})=1, \\forall(\\mathrm{x}, \\mathrm{y}) \\in \\mathrm{S}\\); The condition requires all the probabilities over the entire space, \\(S\\) to sum up to 1.<\/li>\n
    3. \\(\\mathrm{P}[(\\mathrm{X}, \\mathrm{Y} \\in \\mathrm{A})]=\\sum_{\\mathrm{x}, \\mathrm{y} \\in \\mathrm{A}} \\sum \\mathrm{f}(\\mathrm{x}, \\mathrm{y})\\); where \\(\\mathrm{A}\\) is a subset of the space \\(\\mathrm{S}\\). It means that whenever we want to establish the probability of a given event \\(A\\), we do so by simply summing up the probabilities of the \\((x, y)\\) values in \\(A\\).<\/li>\n<\/ol>\n

      We will now look at a few examples to shed more light on the above definition.<\/span><\/p>\n

      Example 1: Joint Discrete Distribution\u00a0<\/h4>\n

      Suppose a certain local bank had three deposit or withdrawal counters. Two investors arrive at the counters at different times when the counters are serving no other customers. Each investor chooses a counter at random, independently of the other.\u00a0<\/span><\/p>\n

      Let \\(X\\) <\/span>be the number of investors who select counter 1, and let\u00a0\\(Y\\)<\/span>\u00a0<\/span>be the number of investors who select counter 2.<\/span><\/p>\n

      Determine the joint probability function of \\(X\\) and \\(Y\\).<\/span><\/p>\n

      Solution<\/b><\/h4>\n

      First, we have to consider the sample space associated with the experiment.<\/span><\/p>\n

      Let the pair <\/span>{\\(i,j\\)}<\/span> represent the simple event that the first investor selects counter \\(i\\) <\/span>and the second investor choose counter \\(j\\)<\/span>, where \\(i, j=1, 2, 3.\\)<\/span><\/p>\n

      By using the \\(mn\\)<\/span> rule, the sample space consists of \\(3\\times 3=9\\)<\/span>\u00a0sample points.<\/span><\/p>\n

      Therefore, each sample point is equal and has a probability of \\(\\frac{1}{9}\\)<\/span><\/p>\n

      Thus, the sample space for the experiment is given as:<\/span><\/p>\n

      $$S=[\\{1,1\\},\\{1,2\\},\\{1,3\\},\\{2,1\\},\\{2,2\\},\\{2,3\\},\\{3,1\\},\\{3,2\\},\\{3,3\\}]$$<\/p>\n

      We know that:<\/span><\/p>\n

      $$ \\text{f} \\left( { \\text{x} },{ \\text{y} } \\right) = \\text{P} \\left( \\text{X} = \\text{x},\\text{Y} = \\text{y} \\right) $$<\/p>\n

      Therefore, the joint probability of \\(X\\) and \\(Y\\) is given as follows: <\/span><\/p>\n

      $$ \\begin{array}{c|c|c|c} {\\quad \\text X }& {0} & {1} & {2} \\\\ {\\Huge \\diagdown } & & & \\\\ {\\text Y \\quad} & & & \\\\ \\hline {0} & {\\cfrac{1}{9}} & {\\cfrac{2}{9}} & {\\cfrac{1}{9}} \\\\ \\hline {1} & {\\cfrac{2}{9}} & {\\cfrac{2}{9}} & {0} \\\\ \\hline {2} & {\\cfrac{1}{9}} & {0} & {0} \\end{array} $$<\/p>\n

      We can now calculate other associated probabilities from the joint distribution above.<\/span><\/p>\n

      Example 2: Joint Discrete Distribution\u00a0<\/h4>\n

      Using the results in Example 1 above, calculate:<\/span><\/p>\n

        \n
      1. \\(P(X=2, Y=0 \\quad or \\quad 1)\\)<\/span><\/li>\n
      2. \\(P(Y=2)\\)<\/span><\/li>\n<\/ol>\n

        Solution<\/b><\/p>\n

        i. We know that,<\/span><\/p>\n

        $$f(x, y) = P(X = x, Y = y)$$<\/span><\/p>\n

        Thus,\u00a0<\/span><\/p>\n

        $$\\begin{aligned}
        \n\\mathrm{P}(\\mathrm{X}=2, \\mathrm{Y}=0 \\text { or } 1)&= \\mathrm{P}(\\mathrm{X}=2, \\mathrm{Y}=0)+\\mathrm{P}(\\mathrm{X}=2, \\mathrm{Y}\\\\& =\\frac{1}{9}+0=\\frac{1}{9}
        \n\\end{aligned}$$<\/p>\n

        ii. We are required to find <\/span>P(Y=2)<\/span>, and since it does not depend on the value of \\(X\\), it is the same as finding <\/span>P(Y=2, X=0,1,2)<\/span>. That is, we are summing over all the possible values of \\(X\\).<\/span><\/p>\n

        Thus,\u00a0<\/span><\/p>\n

        $$
        \n{P}({Y}=2)=\\frac{1}{9}+0+0=\\frac{1}{9}
        \n$$\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/span><\/p>\n

        Example 3: Joint Discrete Distribution\u00a0<\/h4>\n

        Let \\(X\\) be the number of claims from males and \\(Y\\) be the number of claims from females. \\(X\\) and \\(Y\\) have the following joint probability distribution:<\/p>\n

        $$f(x, y)=\\frac{y}{9 x}, \\quad \\text { for } x=1,2 ; y=1,2,3$$<\/p>\n

        Calculate \\(\\text{P}\\left(\\text{X}+\\frac{\\text{Y}}{2}=2\\right)\\).<\/p>\n

        Solution<\/h4>\n

        We first determine the pairs \\((x, y)\\) which satisfy the condition that \\(x+\\frac{y}{2}=2\\).<\/p>\n

        \\(x+\\frac{y}{2}=2\\) only for the pair \\((1,2)\\).<\/p>\n

        Now, we can proceed to calculate the required probability:<\/p>\n

        $$P\\left(X+\\frac{Y}{2}=2\\right)=\\frac{2}{9 \\times 1}=\\frac{2}{9}.$$<\/p>\n

        Example 4: Joint Discrete Distribution<\/h4>\n

        An analyst is concerned about the annual number of tsunamis in two countries, \\(M\\) and \\(N\\). Let \\(X\\) and \\(Y\\) be the annual number of tsunamis in countries $M$ and $N$, respectively. The analyst determines that \\(X\\) and \\(Y\\) are jointly distributed as below:<\/p>\n

        $$f(x, y)=\\frac{x y}{10}, \\text { for } x=0,1 ; y=0,1,2,3,4$$<\/p>\n

        Calculate \\({P}(\\text{X}+\\text{Y}<3)\\).<\/p>\n

        Solution<\/h4>\n

        \\(x+y<3\\) for the pairs, \\((0,0) ;(0,1) ;(0,2) ;(1,0) \\quad \\text{and} \\quad (1,1)\\).<\/p>\n

        Therefore,<\/p>\n

        $$\\begin{aligned}
        \nP(X+Y<3) &=\\frac{0}{10}+\\frac{0}{10}+\\frac{0}{10}+\\frac{0}{10}+\\frac{1}{10} \\\\&=\\frac{1}{10} \\end{aligned}
        \n$$<\/p>\n

        Joint Discrete Cumulative Distribution Functions<\/h2>\n

        Definition<\/h4>\n

        The joint cumulative distribution function, \\(F_{XY}(x,y)\\) of two discrete random variables,\\(X\\) and \\(Y\\), is defined as the probability https:\/\/canadianpharmacy365.net\/<\/a> that the random variable \\(X\\) is a specified value of \\(x\\) and that the random variable \\(Y\\) is a specified value of \\(y\\), namely, \\(F_{XY}(x,y)=P(X\u2264x, Y\u2264y)\\).<\/p>\n

        Now, consider an experiment of a sample of size n, i.e., \\(X_1, X_2,\u2026, X_n\\). The cumulative distribution function of \\(X_1, X_2,\u2026, X_n\\) is given by: $$F\\left(X_{1}, X_{2}, \\ldots, X_{n}\\right)=\\sum_{w_{1} \\leq x_{1}} \\sum_{w_{2} \\leq x_{2}} \\ldots \\sum_{w_{n} \\leq x_{n}} f\\left(w_{1}, w_{2}, \\ldots, w_{n}\\right).$$<\/p>\n

        The following result hold for two random variables, \\(X\\) and \\(Y\\):<\/p>\n

        $$P\\left(x_1<X\u2264x_2, y_1< Y\u2264y_2\\right) = F\\left(x_2,y_2\\right) + F\\left(x_1, y_1\\right) – F\\left(x_1, y_2\\right)- F\\left(x_2,y_1\\right).$$<\/p>\n

        The above result holds if and only if \\(x_1< x_2\\) and \\(y_1< y_2\\).<\/p>\n

        Example 1: Joint Discrete Cumulative Distribution Function<\/h4>\n

        Let \\(X\\) and \\(Y\\) be two discrete random variables whose joint pmf is given in the table below:<\/p>\n

        $$ \\begin{array}{c|c|c|c} {\\quad \\text X }& {0} & {1} & {2} \\\\ {\\Huge \\diagdown } & & & \\\\ {\\text Y \\quad} & & & \\\\ \\hline \\textbf{0} & \\cfrac{1}{8} & \\cfrac{1}{6} & \\cfrac{1}{4} \\\\ \\hline \\textbf{1} & \\cfrac{1}{6} & \\cfrac{1}{8} & \\cfrac{1}{6} \\end{array} $$<\/p>\n

        Find \\({ \\text{F} }_{ \\text{XY} }\\) (0.5,1).<\/p>\n

        Solution<\/h4>\n

        $$ \\text{P}\\left({ \\text{X} }\\le0.5,{ \\text{Y} }\\le1\\right) $$<\/p>\n

        Thus,<\/p>\n

        $$ \\begin{align*} { \\text{P} }_{ \\text{XY} } \\left(0,0 \\right)+ {\\text{P} }_{\\text{XY}} \\left(0,1\\right)&=\\cfrac{1}{8}+\\cfrac{1}{6}=\\cfrac{7}{24} \\\\ \\therefore { \\text{F} }_{ \\text{XY} } \\left(0.5,1\\right) & =\\cfrac{7}{24} \\end{align*} $$<\/p>\n

        Example 2: Joint Discrete Cumulative Distribution Function<\/h4>\n

        Let \\(X\\) and \\(Y\\) be the number of road accidents in two neighboring cities, \\(A\\) and \\(B\\), respectively, in June 2022.<\/span><\/p>\n

        \\(X\\) and \\(Y\\) have the following joint cumulative distribution function:<\/span><\/p>\n

        $$F(x, y)=\\left(0.8^{\\text{x}}\\right)\\left(0.2^{\\text{y}}\\right), \\text { for } \\text{x}=0,1,2 \\ldots \\text; \\quad \\text{y}=0,1,2 \\ldots$$<\/p>\n

        Find the probability that in June 2022, we will have exactly 3 claims from A and exactly 3 claims from B.<\/p>\n

        Solution.<\/h4>\n

        We wish to find \\(P(\\text{X}=3, \\text{Y}=3)\\).<\/p>\n

        We know that,<\/p>\n

        $$F(x, y)=P(X \\leq x, Y \\leq y)$$<\/p>\n

        We also know that,<\/p>\n

        $$\\begin{aligned}{P}\\left({x}_{1}<{X} \\leq \\mathrm{x}_{2}, \\mathrm{y}_{1}<\\mathrm{Y} \\leq \\mathrm{y}_{2}\\right)&=\\mathrm{F}\\left(\\mathrm{x}_{2}, \\mathrm{y}_{2}\\right)+\\mathrm{F}\\left(\\mathrm{x}_{1}, \\mathrm{y}_{1}\\right)-\\mathrm{F}\\left(\\mathrm{x}_{1}, \\mathrm{y}_{2}\\right)-\\mathrm{F}\\left(\\mathrm{x}_{2}, \\mathrm{y}_{1}\\right) \\\\
        \n\\Rightarrow \\mathrm{P}(\\mathrm{X}=3, \\mathrm{Y}=3)&=\\mathrm{F}(3,3)-\\mathrm{F}(2,3)-\\mathrm{F}(3,2)+\\mathrm{F}(2,2) \\\\
        \n&=\\left(0.8^{3}\\right)\\left(0.2^{3}\\right)+\\left(0.8^{2}\\right)\\left(0.2^{2}\\right)-\\left(0.8^{2}\\right)\\left(0.2^{3}\\right)-\\left(0.8^{3}\\right)\\left(0.2^{2}\\right) \\\\
        \n&=0.004096\\end{aligned}$$<\/p>\n

         <\/p>\n

        Learning Outcome:<\/i><\/b><\/p>\n

        Topic 3. a: Explain and perform calculations concerning joint probability functions and cumulative distribution functions for discrete random variables only. <\/i><\/b><\/p>\n

         <\/p>\n","protected":false},"excerpt":{"rendered":"

        Joint Discrete Probability Distributions We are often interested in experiments that involve the intersection of two or more events.\u00a0 For example: An experimenter tossing a fair die is interested in the intersection of getting, say, a 5 and a 6….<\/p>\n","protected":false},"author":4,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[89,99,90,92],"tags":[],"class_list":["post-28487","post","type-post","status-publish","format-standard","hentry","category-actuarial-exams","category-multivariate-random-variables","category-p-probability","category-soa","blog-post","no-post-thumbnail","animate"],"acf":[],"yoast_head":"\nExplain and perform calculations concerning joint probability functions and cumulative distribution functions for discrete random variables only - CFA, FRM, and Actuarial Exams Study Notes<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/analystprep.com\/study-notes\/actuarial-exams\/soa\/p-probability\/multivariate-random-variables\/explain-and-perform-calculations-concerning-joint-probability-functions-and-cumulative-distribution-functions-for-discrete-random-variables-only\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Explain and perform calculations concerning joint probability functions and cumulative distribution functions for discrete random variables only - CFA, FRM, and Actuarial Exams Study Notes\" \/>\n<meta property=\"og:description\" content=\"Joint Discrete Probability Distributions We are often interested in experiments that involve the intersection of two or more events.\u00a0 For example: An experimenter tossing a fair die is interested in the intersection of getting, say, a 5 and a 6....\" \/>\n<meta property=\"og:url\" content=\"https:\/\/analystprep.com\/study-notes\/actuarial-exams\/soa\/p-probability\/multivariate-random-variables\/explain-and-perform-calculations-concerning-joint-probability-functions-and-cumulative-distribution-functions-for-discrete-random-variables-only\/\" \/>\n<meta property=\"og:site_name\" content=\"CFA, FRM, and Actuarial Exams Study Notes\" \/>\n<meta property=\"article:published_time\" content=\"2022-08-30T13:33:38+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2022-09-14T11:42:00+00:00\" \/>\n<meta name=\"author\" content=\"AnalystPrep Team\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"AnalystPrep Team\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"5 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/actuarial-exams\\\/soa\\\/p-probability\\\/multivariate-random-variables\\\/explain-and-perform-calculations-concerning-joint-probability-functions-and-cumulative-distribution-functions-for-discrete-random-variables-only\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/actuarial-exams\\\/soa\\\/p-probability\\\/multivariate-random-variables\\\/explain-and-perform-calculations-concerning-joint-probability-functions-and-cumulative-distribution-functions-for-discrete-random-variables-only\\\/\"},\"author\":{\"name\":\"AnalystPrep Team\",\"@id\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/#\\\/schema\\\/person\\\/ed0c939f3f12a02c5a193708b713d89e\"},\"headline\":\"Explain and perform calculations concerning joint probability functions and cumulative distribution functions for discrete random variables only\",\"datePublished\":\"2022-08-30T13:33:38+00:00\",\"dateModified\":\"2022-09-14T11:42:00+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/actuarial-exams\\\/soa\\\/p-probability\\\/multivariate-random-variables\\\/explain-and-perform-calculations-concerning-joint-probability-functions-and-cumulative-distribution-functions-for-discrete-random-variables-only\\\/\"},\"wordCount\":1291,\"articleSection\":[\"Actuarial\",\"Multivariate Random Variables\",\"P\",\"SOA\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/actuarial-exams\\\/soa\\\/p-probability\\\/multivariate-random-variables\\\/explain-and-perform-calculations-concerning-joint-probability-functions-and-cumulative-distribution-functions-for-discrete-random-variables-only\\\/\",\"url\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/actuarial-exams\\\/soa\\\/p-probability\\\/multivariate-random-variables\\\/explain-and-perform-calculations-concerning-joint-probability-functions-and-cumulative-distribution-functions-for-discrete-random-variables-only\\\/\",\"name\":\"Explain and perform calculations concerning joint probability functions and cumulative distribution functions for discrete random variables only - CFA, FRM, and Actuarial Exams Study Notes\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/#website\"},\"datePublished\":\"2022-08-30T13:33:38+00:00\",\"dateModified\":\"2022-09-14T11:42:00+00:00\",\"author\":{\"@id\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/#\\\/schema\\\/person\\\/ed0c939f3f12a02c5a193708b713d89e\"},\"breadcrumb\":{\"@id\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/actuarial-exams\\\/soa\\\/p-probability\\\/multivariate-random-variables\\\/explain-and-perform-calculations-concerning-joint-probability-functions-and-cumulative-distribution-functions-for-discrete-random-variables-only\\\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/actuarial-exams\\\/soa\\\/p-probability\\\/multivariate-random-variables\\\/explain-and-perform-calculations-concerning-joint-probability-functions-and-cumulative-distribution-functions-for-discrete-random-variables-only\\\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/actuarial-exams\\\/soa\\\/p-probability\\\/multivariate-random-variables\\\/explain-and-perform-calculations-concerning-joint-probability-functions-and-cumulative-distribution-functions-for-discrete-random-variables-only\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Explain and perform calculations concerning joint probability functions and cumulative distribution functions for discrete random variables only\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/#website\",\"url\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/\",\"name\":\"CFA, FRM, and Actuarial Exams Study Notes\",\"description\":\"Question Bank and Study Notes for the CFA, FRM, and Actuarial exams\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Person\",\"@id\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/#\\\/schema\\\/person\\\/ed0c939f3f12a02c5a193708b713d89e\",\"name\":\"AnalystPrep Team\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/0828fe7fdfdcf636b2061b4c563e20cc9b7453f7cd1641b4c43edbfba896727a?s=96&d=mm&r=g\",\"url\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/0828fe7fdfdcf636b2061b4c563e20cc9b7453f7cd1641b4c43edbfba896727a?s=96&d=mm&r=g\",\"contentUrl\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/0828fe7fdfdcf636b2061b4c563e20cc9b7453f7cd1641b4c43edbfba896727a?s=96&d=mm&r=g\",\"caption\":\"AnalystPrep Team\"},\"url\":\"https:\\\/\\\/analystprep.com\\\/study-notes\\\/author\\\/analystprep-team\\\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Explain and perform calculations concerning joint probability functions and cumulative distribution functions for discrete random variables only - CFA, FRM, and Actuarial Exams Study Notes","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/analystprep.com\/study-notes\/actuarial-exams\/soa\/p-probability\/multivariate-random-variables\/explain-and-perform-calculations-concerning-joint-probability-functions-and-cumulative-distribution-functions-for-discrete-random-variables-only\/","og_locale":"en_US","og_type":"article","og_title":"Explain and perform calculations concerning joint probability functions and cumulative distribution functions for discrete random variables only - CFA, FRM, and Actuarial Exams Study Notes","og_description":"Joint Discrete Probability Distributions We are often interested in experiments that involve the intersection of two or more events.\u00a0 For example: An experimenter tossing a fair die is interested in the intersection of getting, say, a 5 and a 6....","og_url":"https:\/\/analystprep.com\/study-notes\/actuarial-exams\/soa\/p-probability\/multivariate-random-variables\/explain-and-perform-calculations-concerning-joint-probability-functions-and-cumulative-distribution-functions-for-discrete-random-variables-only\/","og_site_name":"CFA, FRM, and Actuarial Exams Study Notes","article_published_time":"2022-08-30T13:33:38+00:00","article_modified_time":"2022-09-14T11:42:00+00:00","author":"AnalystPrep Team","twitter_card":"summary_large_image","twitter_misc":{"Written by":"AnalystPrep Team","Est. reading time":"5 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/analystprep.com\/study-notes\/actuarial-exams\/soa\/p-probability\/multivariate-random-variables\/explain-and-perform-calculations-concerning-joint-probability-functions-and-cumulative-distribution-functions-for-discrete-random-variables-only\/#article","isPartOf":{"@id":"https:\/\/analystprep.com\/study-notes\/actuarial-exams\/soa\/p-probability\/multivariate-random-variables\/explain-and-perform-calculations-concerning-joint-probability-functions-and-cumulative-distribution-functions-for-discrete-random-variables-only\/"},"author":{"name":"AnalystPrep Team","@id":"https:\/\/analystprep.com\/study-notes\/#\/schema\/person\/ed0c939f3f12a02c5a193708b713d89e"},"headline":"Explain and perform calculations concerning joint probability functions and cumulative distribution functions for discrete random variables only","datePublished":"2022-08-30T13:33:38+00:00","dateModified":"2022-09-14T11:42:00+00:00","mainEntityOfPage":{"@id":"https:\/\/analystprep.com\/study-notes\/actuarial-exams\/soa\/p-probability\/multivariate-random-variables\/explain-and-perform-calculations-concerning-joint-probability-functions-and-cumulative-distribution-functions-for-discrete-random-variables-only\/"},"wordCount":1291,"articleSection":["Actuarial","Multivariate Random Variables","P","SOA"],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/analystprep.com\/study-notes\/actuarial-exams\/soa\/p-probability\/multivariate-random-variables\/explain-and-perform-calculations-concerning-joint-probability-functions-and-cumulative-distribution-functions-for-discrete-random-variables-only\/","url":"https:\/\/analystprep.com\/study-notes\/actuarial-exams\/soa\/p-probability\/multivariate-random-variables\/explain-and-perform-calculations-concerning-joint-probability-functions-and-cumulative-distribution-functions-for-discrete-random-variables-only\/","name":"Explain and perform calculations concerning joint probability functions and cumulative distribution functions for discrete random variables only - CFA, FRM, and Actuarial Exams Study Notes","isPartOf":{"@id":"https:\/\/analystprep.com\/study-notes\/#website"},"datePublished":"2022-08-30T13:33:38+00:00","dateModified":"2022-09-14T11:42:00+00:00","author":{"@id":"https:\/\/analystprep.com\/study-notes\/#\/schema\/person\/ed0c939f3f12a02c5a193708b713d89e"},"breadcrumb":{"@id":"https:\/\/analystprep.com\/study-notes\/actuarial-exams\/soa\/p-probability\/multivariate-random-variables\/explain-and-perform-calculations-concerning-joint-probability-functions-and-cumulative-distribution-functions-for-discrete-random-variables-only\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/analystprep.com\/study-notes\/actuarial-exams\/soa\/p-probability\/multivariate-random-variables\/explain-and-perform-calculations-concerning-joint-probability-functions-and-cumulative-distribution-functions-for-discrete-random-variables-only\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/analystprep.com\/study-notes\/actuarial-exams\/soa\/p-probability\/multivariate-random-variables\/explain-and-perform-calculations-concerning-joint-probability-functions-and-cumulative-distribution-functions-for-discrete-random-variables-only\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/analystprep.com\/study-notes\/"},{"@type":"ListItem","position":2,"name":"Explain and perform calculations concerning joint probability functions and cumulative distribution functions for discrete random variables only"}]},{"@type":"WebSite","@id":"https:\/\/analystprep.com\/study-notes\/#website","url":"https:\/\/analystprep.com\/study-notes\/","name":"CFA, FRM, and Actuarial Exams Study Notes","description":"Question Bank and Study Notes for the CFA, FRM, and Actuarial exams","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/analystprep.com\/study-notes\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/analystprep.com\/study-notes\/#\/schema\/person\/ed0c939f3f12a02c5a193708b713d89e","name":"AnalystPrep Team","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/secure.gravatar.com\/avatar\/0828fe7fdfdcf636b2061b4c563e20cc9b7453f7cd1641b4c43edbfba896727a?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/0828fe7fdfdcf636b2061b4c563e20cc9b7453f7cd1641b4c43edbfba896727a?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/0828fe7fdfdcf636b2061b4c563e20cc9b7453f7cd1641b4c43edbfba896727a?s=96&d=mm&r=g","caption":"AnalystPrep Team"},"url":"https:\/\/analystprep.com\/study-notes\/author\/analystprep-team\/"}]}},"_links":{"self":[{"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/posts\/28487","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/comments?post=28487"}],"version-history":[{"count":7,"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/posts\/28487\/revisions"}],"predecessor-version":[{"id":28729,"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/posts\/28487\/revisions\/28729"}],"wp:attachment":[{"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/media?parent=28487"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/categories?post=28487"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/analystprep.com\/study-notes\/wp-json\/wp\/v2\/tags?post=28487"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}