{"id":246,"date":"2021-07-14T15:58:58","date_gmt":"2021-07-14T15:58:58","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/poisson-distribution\/"},"modified":"2023-12-05T09:11:46","modified_gmt":"2023-12-05T09:11:46","slug":"poisson-distribution","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/poisson-distribution\/","title":{"raw":"What are Poisson Experiments?","rendered":"What are Poisson Experiments?"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<section>\r\n<ul id=\"list14235\">\r\n \t<li>Describe the two characteristics of a poisson experiment<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\nThere are two main characteristics of a Poisson experiment.\r\n<ol>\r\n \t<li>The <strong>Poisson probability distribution<\/strong> gives the probability of a number of events occurring in a <strong>fixed interval<\/strong> of time or space if these events happen with a known average rate and independently of the time since the last event. For example, a book editor might be interested in the number of words spelled incorrectly in a particular book. It might be that, on the average, there are five words spelled incorrectly in 100 pages. The interval is the 100 pages.<\/li>\r\n \t<li>The Poisson distribution may be used to approximate the binomial if the probability of success is \"small\" (such as 0.01) and the number of trials is \"large\" (such as 1,000). You will verify the relationship in the homework exercises. [latex]n[\/latex] is the number of trials, and [latex]p[\/latex]\u00a0is the probability of a \"success.\"<\/li>\r\n<\/ol>\r\nThe random variable [latex]X=[\/latex] the number of occurrences in the interval of interest.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nThe average number of loaves of bread put on a shelf in a bakery in a half-hour period is 12. Of interest is the number of loaves of bread put on the shelf in five minutes. The time interval of interest is five minutes. What is the probability that the number of loaves, selected randomly, put on the shelf in five minutes is three?\r\n\r\n[reveal-answer q=\"869903\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"869903\"]\r\n\r\nLet [latex]X=[\/latex] the number of loaves of bread put on the shelf in five minutes. If the average number of loaves put on the shelf in 30 minutes (half-hour) is 12, <strong>then the average number of loaves put on the shelf in five minutes is<\/strong>\u00a0[latex]\\left(\\frac{5}{30}\\right)\\left(12\\right)=2[\/latex]\u00a0loaves of bread.\r\n\r\nThe probability question asks you to find [latex]P(x=3)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nThe average number of fish caught in an hour is eight. Of interest is the number of fish caught in 15 minutes. The time interval of interest is 15 minutes. What is the average number of fish caught in 15 minutes?\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nA bank expects to receive six bad checks per day, on average. What is the probability of the bank getting fewer than five bad checks on any given day? Of interest is the number of checks the bank receives in one day, so the time interval of interest is one day. Let\u00a0[latex]X[\/latex] = the number of bad checks the bank receives in one day. If the bank expects to receive six bad checks per day then the average is six checks per day. Write a mathematical statement for the probability question.\r\n\r\n[reveal-answer q=\"609883\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"609883\"]\r\n\r\n[latex]P(x&lt;5)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nAn electronics store expects to have ten returns per day on average. The manager wants to know the probability of the store getting fewer than eight returns on any given day. State the probability question mathematically.\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<p id=\"element-234\" class=\" \">You notice that a news reporter says \"uh,\" on average, two times per broadcast. What is the probability that the news reporter says \"uh\" more than two times per broadcast.<\/p>\r\n<p id=\"element-235\" class=\" \">This is a Poisson problem because you are interested in knowing the number of times the news reporter says \"uh\" during a broadcast.<\/p>\r\n\r\n<div id=\"element-895\" class=\" unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"id18394368\" data-type=\"problem\">\r\n<div class=\"os-problem-container \">\r\n<p id=\"element-734\" class=\" \">a. What is the interval of interest?<\/p>\r\n<p id=\"element-10\" class=\" \">b. What is the average number of times the news reporter says \"uh\" during one broadcast?<\/p>\r\n<p id=\"element-817\" class=\" \">c. Let\u00a0[latex]X[\/latex]\u00a0= ____________. What values does\u00a0[latex]X[\/latex]\u00a0take on?<\/p>\r\n<p id=\"element-423\" class=\" \">d. The probability question is [latex]P[\/latex](______).<\/p>\r\n[reveal-answer q=\"438880\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"438880\"]\r\n<p id=\"element-227\" class=\" \">a. one broadcast<\/p>\r\n<p id=\"element-40\" class=\" \">b. 2<\/p>\r\n<p id=\"element-112\" class=\" \">c. Let\u00a0[latex]X[\/latex]\u00a0= the number of times the news reporter says \"uh\" during one broadcast.\r\n[latex]x[\/latex]\u00a0= 0, 1, 2, 3, ...<\/p>\r\n<p id=\"element-683\" class=\" \">d. [latex]P(x&gt;2)[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nAn emergency room at a particular hospital gets an average of five patients per hour. A doctor wants to know the probability that the ER gets more than five patients per hour. Give the reason why this would be a Poisson distribution.\r\n\r\n<\/div>\r\n&nbsp;","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<section>\n<ul id=\"list14235\">\n<li>Describe the two characteristics of a poisson experiment<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<p>There are two main characteristics of a Poisson experiment.<\/p>\n<ol>\n<li>The <strong>Poisson probability distribution<\/strong> gives the probability of a number of events occurring in a <strong>fixed interval<\/strong> of time or space if these events happen with a known average rate and independently of the time since the last event. For example, a book editor might be interested in the number of words spelled incorrectly in a particular book. It might be that, on the average, there are five words spelled incorrectly in 100 pages. The interval is the 100 pages.<\/li>\n<li>The Poisson distribution may be used to approximate the binomial if the probability of success is &#8220;small&#8221; (such as 0.01) and the number of trials is &#8220;large&#8221; (such as 1,000). You will verify the relationship in the homework exercises. [latex]n[\/latex] is the number of trials, and [latex]p[\/latex]\u00a0is the probability of a &#8220;success.&#8221;<\/li>\n<\/ol>\n<p>The random variable [latex]X=[\/latex] the number of occurrences in the interval of interest.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>The average number of loaves of bread put on a shelf in a bakery in a half-hour period is 12. Of interest is the number of loaves of bread put on the shelf in five minutes. The time interval of interest is five minutes. What is the probability that the number of loaves, selected randomly, put on the shelf in five minutes is three?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q869903\">Show Solution<\/span><\/p>\n<div id=\"q869903\" class=\"hidden-answer\" style=\"display: none\">\n<p>Let [latex]X=[\/latex] the number of loaves of bread put on the shelf in five minutes. If the average number of loaves put on the shelf in 30 minutes (half-hour) is 12, <strong>then the average number of loaves put on the shelf in five minutes is<\/strong>\u00a0[latex]\\left(\\frac{5}{30}\\right)\\left(12\\right)=2[\/latex]\u00a0loaves of bread.<\/p>\n<p>The probability question asks you to find [latex]P(x=3)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>The average number of fish caught in an hour is eight. Of interest is the number of fish caught in 15 minutes. The time interval of interest is 15 minutes. What is the average number of fish caught in 15 minutes?<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>A bank expects to receive six bad checks per day, on average. What is the probability of the bank getting fewer than five bad checks on any given day? Of interest is the number of checks the bank receives in one day, so the time interval of interest is one day. Let\u00a0[latex]X[\/latex] = the number of bad checks the bank receives in one day. If the bank expects to receive six bad checks per day then the average is six checks per day. Write a mathematical statement for the probability question.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q609883\">Show Answer<\/span><\/p>\n<div id=\"q609883\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]P(x<5)[\/latex]\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>An electronics store expects to have ten returns per day on average. The manager wants to know the probability of the store getting fewer than eight returns on any given day. State the probability question mathematically.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p id=\"element-234\" class=\"\">You notice that a news reporter says &#8220;uh,&#8221; on average, two times per broadcast. What is the probability that the news reporter says &#8220;uh&#8221; more than two times per broadcast.<\/p>\n<p id=\"element-235\" class=\"\">This is a Poisson problem because you are interested in knowing the number of times the news reporter says &#8220;uh&#8221; during a broadcast.<\/p>\n<div id=\"element-895\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"id18394368\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"element-734\" class=\"\">a. What is the interval of interest?<\/p>\n<p id=\"element-10\" class=\"\">b. What is the average number of times the news reporter says &#8220;uh&#8221; during one broadcast?<\/p>\n<p id=\"element-817\" class=\"\">c. Let\u00a0[latex]X[\/latex]\u00a0= ____________. What values does\u00a0[latex]X[\/latex]\u00a0take on?<\/p>\n<p id=\"element-423\" class=\"\">d. The probability question is [latex]P[\/latex](______).<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q438880\">Show Answer<\/span><\/p>\n<div id=\"q438880\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"element-227\" class=\"\">a. one broadcast<\/p>\n<p id=\"element-40\" class=\"\">b. 2<\/p>\n<p id=\"element-112\" class=\"\">c. Let\u00a0[latex]X[\/latex]\u00a0= the number of times the news reporter says &#8220;uh&#8221; during one broadcast.<br \/>\n[latex]x[\/latex]\u00a0= 0, 1, 2, 3, &#8230;<\/p>\n<p id=\"element-683\" class=\"\">d. [latex]P(x>2)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>An emergency room at a particular hospital gets an average of five patients per hour. A doctor wants to know the probability that the ER gets more than five patients per hour. Give the reason why this would be a Poisson distribution.<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-246\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>The Poisson Distribution. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/4-6-poisson-distribution\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/4-6-poisson-distribution<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/li><li>Introductory Statistics. <strong>Authored by<\/strong>: Barbara Illowsky, Susan Dean. <strong>Provided by<\/strong>: Open Stax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169134,"menu_order":26,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"The Poisson Distribution\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/4-6-poisson-distribution\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"},{\"type\":\"cc\",\"description\":\"Introductory Statistics\",\"author\":\"Barbara Illowsky, Susan Dean\",\"organization\":\"Open Stax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-246","chapter","type-chapter","status-publish","hentry"],"part":240,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/246","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/users\/169134"}],"version-history":[{"count":15,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/246\/revisions"}],"predecessor-version":[{"id":3596,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/246\/revisions\/3596"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/240"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/246\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=246"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=246"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=246"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=246"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}