{"id":537,"date":"2020-07-18T19:00:17","date_gmt":"2020-07-18T23:00:17","guid":{"rendered":"https:\/\/openbooks.macewan.ca\/rcommander\/?post_type=chapter&p=537"},"modified":"2024-06-06T18:08:03","modified_gmt":"2024-06-06T22:08:03","slug":"4-1-random-variable","status":"publish","type":"chapter","link":"https:\/\/openbooks.macewan.ca\/introstats\/chapter\/4-1-random-variable\/","title":{"raw":"4.1 Random Variable","rendered":"4.1 Random Variable"},"content":{"raw":"In this section we introduce discrete random variables and their probability distributions.\r\n\r\nGiven a chance experiment, the collection of possible outcomes is called the sample space, denoted as S<\/em><\/strong>. A random variable <\/strong>is a function (or a mapping) from the sample space S<\/em><\/strong> into real numbers. Random variables are usually denoted as uppercase letters, such as X, Y, Z. We use the corresponding lowercase letters x, y, z to represent possible values that random variables may attain.\r\n
\r\n

Example: Random Variable<\/p>\r\n\r\n<\/header>\r\n

\r\n
    \r\n \t
  1. Consider the chance experiment of flipping a balanced coin twice, the sample space is S <\/em><\/strong>= {HH, HT, TT, TH}. Let the random variable X = # of tails. It is a mapping from the sample space S<\/em><\/strong> to integers 0, 1, and 2.<\/a>[caption id=\"attachment_3552\" align=\"aligncenter\" width=\"326\"]\"Two Figure 4.1<\/strong>: Random Variable X=# of Tails. [Image Description (See Appendix D Figure 4.1)<\/a>][\/caption]<\/li>\r\n \t
  2. Five students are asked to report the number of siblings they have; their responses are summarized in the following table:\r\n
    \r\n\r\n\r\n\r\n\r\n
    Name<\/th>\r\nMark<\/td>\r\nJohn<\/td>\r\nRebecca<\/td>\r\nSarah<\/td>\r\nMary<\/td>\r\n<\/tr>\r\n
    # of Siblings<\/th>\r\n0<\/td>\r\n1<\/td>\r\n2<\/td>\r\n2<\/td>\r\n3<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nRandomly pick one student and let random variable X = # of siblings the student has. Then X is a mapping from the sample space S<\/em><\/strong> = {Mark, John, Rebecca, Sarah, Mary} to the numbers 0, 1, 2 and 3.<\/a><\/li>\r\n<\/ol>\r\n[caption id=\"attachment_1629\" align=\"aligncenter\" width=\"326\"]\"Two Figure 4.2<\/strong>: Random Variable X=# of Siblings. [Image Description (See Appendix D Figure 4.2)<\/a>][\/caption]In general, a discrete random variable maps the sample space S<\/em><\/strong> to numbers that can be listed or counted; a continuous random variable maps the sample space S<\/em><\/strong> to an interval that is a subset of the entire real line. If you need to review discrete and continuous data, refer to variables and data<\/a> in Chapter 1.3.<\/div>\r\n<\/div>","rendered":"

    In this section we introduce discrete random variables and their probability distributions.<\/p>\n

    Given a chance experiment, the collection of possible outcomes is called the sample space, denoted as S<\/em><\/strong>. A random variable <\/strong>is a function (or a mapping) from the sample space S<\/em><\/strong> into real numbers. Random variables are usually denoted as uppercase letters, such as X, Y, Z. We use the corresponding lowercase letters x, y, z to represent possible values that random variables may attain.<\/p>\n

    \n
    \n

    Example: Random Variable<\/p>\n<\/header>\n

    \n
      \n
    1. Consider the chance experiment of flipping a balanced coin twice, the sample space is S <\/em><\/strong>= {HH, HT, TT, TH}. Let the random variable X = # of tails. It is a mapping from the sample space S<\/em><\/strong> to integers 0, 1, and 2.<\/a>
      \n
      \"Two
      Figure 4.1<\/strong>: Random Variable X=# of Tails. [Image Description (See Appendix D Figure 4.1)<\/a>]<\/figcaption><\/figure>\n<\/li>\n
    2. Five students are asked to report the number of siblings they have; their responses are summarized in the following table:\n
      \n\n\n\n\n
      Name<\/th>\nMark<\/td>\nJohn<\/td>\nRebecca<\/td>\nSarah<\/td>\nMary<\/td>\n<\/tr>\n
      # of Siblings<\/th>\n0<\/td>\n1<\/td>\n2<\/td>\n2<\/td>\n3<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

      Randomly pick one student and let random variable X = # of siblings the student has. Then X is a mapping from the sample space S<\/em><\/strong> = {Mark, John, Rebecca, Sarah, Mary} to the numbers 0, 1, 2 and 3.<\/a><\/li>\n<\/ol>\n

      \"Two
      Figure 4.2<\/strong>: Random Variable X=# of Siblings. [Image Description (See Appendix D Figure 4.2)<\/a>]<\/figcaption><\/figure>\n

      In general, a discrete random variable maps the sample space S<\/em><\/strong> to numbers that can be listed or counted; a continuous random variable maps the sample space S<\/em><\/strong> to an interval that is a subset of the entire real line. If you need to review discrete and continuous data, refer to variables and data<\/a> in Chapter 1.3.<\/div>\n<\/div>\n","protected":false},"author":19,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-537","chapter","type-chapter","status-publish","hentry"],"part":535,"_links":{"self":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/537","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/users\/19"}],"version-history":[{"count":78,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/537\/revisions"}],"predecessor-version":[{"id":5381,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/537\/revisions\/5381"}],"part":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/parts\/535"}],"metadata":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapters\/537\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/media?parent=537"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/pressbooks\/v2\/chapter-type?post=537"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/contributor?post=537"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/openbooks.macewan.ca\/introstats\/wp-json\/wp\/v2\/license?post=537"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}