{"id":1023,"date":"2023-06-22T01:45:08","date_gmt":"2023-06-22T01:45:08","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/probability-distributions-apply-it-3\/"},"modified":"2025-05-16T02:19:28","modified_gmt":"2025-05-16T02:19:28","slug":"probability-distributions-apply-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/introstatstest\/chapter\/probability-distributions-apply-it-3\/","title":{"raw":"Probability Distributions: Apply It 3","rendered":"Probability Distributions: Apply It 3"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Understand the concept of a probability distribution and its role in describing the behavior of a random variable.<\/li>\r\n\t<li>Describe the characteristics of probability distributions.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Continuous Probability Distributions<\/h2>\r\n<p>So far, we have been using a discrete probability distribution that gives the probabilities for a fixed set of values. The spinner can land on purple [latex]2[\/latex] or [latex]3[\/latex] times, but [latex]2.5[\/latex] is impossible. It is not in the set of possible values.<\/p>\r\n<p>However, some variables are <strong>continuous<\/strong>, which means the range of values includes an infinite number of possible values. Consider a person\u2019s height. Although we often measure heights to the nearest inch, a person does not grow in one-inch spurts but instead moves through the range of heights via immeasurably small increments. It is not possible to count all the possible heights that a person can be because even between [latex]64[\/latex] inches and [latex]65[\/latex] inches, there are infinitely many possible heights.<\/p>\r\n<p>When we are using a <strong>discrete<\/strong> probability distribution, we calculate the probability for a range of values by adding up the probability of each outcome in the range. However, when we are using a <strong>continuous<\/strong> probability distribution, probabilities are represented as the area under a density curve. The total area under the curve is equal to [latex]1[\/latex].<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>continuous probability distribution<\/h3>\r\n<p>A continuous probability distribution is a probability distribution for a continuous random variable (an infinite and uncountable random variable).<\/p>\r\n<p>&nbsp;<\/p>\r\n<p>The probabilities of a continuous probability distribution are represented as the area under a density curve. The total area under the curve is equal to [latex]1[\/latex].<\/p>\r\n<p>&nbsp;<\/p>\r\n<p><img class=\"aligncenter wp-image-5756 size-medium\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/22014507\/ChisqDist-1.png\" alt=\"A chi-squared distribution with df=3.\" width=\"300\" height=\"178\" \/><img class=\"aligncenter wp-image-5754\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/11\/03191851\/picture.png\" alt=\"The graph of a normal distribution with a mean of 0 and a standard deviation of 1.\" width=\"324\" height=\"175\" \/><\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Suppose the following graph shows the distribution of heights (in inches) for women in a particular country. To find the probability that a randomly selected woman is between 60 and 66 inches tall, we would shade the area under the curve between these two values.<img class=\"aligncenter wp-image-1467 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/10\/06200143\/Picture13.png\" alt=\"A graph of women's height in inches. 76.06% of the heights are between 60 and 66 inches.\" width=\"936\" height=\"362\" \/><\/section>\r\n<p>In some situations, we may use a continuous probability distribution as an approximation even when the variable is technically discrete.<\/p>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]1606[\/ohm2_question]<\/section>\r\n<section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm2_question hide_question_numbers=1]1516[\/ohm2_question]<\/p>\r\n<\/section>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Understand the concept of a probability distribution and its role in describing the behavior of a random variable.<\/li>\n<li>Describe the characteristics of probability distributions.<\/li>\n<\/ul>\n<\/section>\n<h2>Continuous Probability Distributions<\/h2>\n<p>So far, we have been using a discrete probability distribution that gives the probabilities for a fixed set of values. The spinner can land on purple [latex]2[\/latex] or [latex]3[\/latex] times, but [latex]2.5[\/latex] is impossible. It is not in the set of possible values.<\/p>\n<p>However, some variables are <strong>continuous<\/strong>, which means the range of values includes an infinite number of possible values. Consider a person\u2019s height. Although we often measure heights to the nearest inch, a person does not grow in one-inch spurts but instead moves through the range of heights via immeasurably small increments. It is not possible to count all the possible heights that a person can be because even between [latex]64[\/latex] inches and [latex]65[\/latex] inches, there are infinitely many possible heights.<\/p>\n<p>When we are using a <strong>discrete<\/strong> probability distribution, we calculate the probability for a range of values by adding up the probability of each outcome in the range. However, when we are using a <strong>continuous<\/strong> probability distribution, probabilities are represented as the area under a density curve. The total area under the curve is equal to [latex]1[\/latex].<\/p>\n<section class=\"textbox keyTakeaway\">\n<h3>continuous probability distribution<\/h3>\n<p>A continuous probability distribution is a probability distribution for a continuous random variable (an infinite and uncountable random variable).<\/p>\n<p>&nbsp;<\/p>\n<p>The probabilities of a continuous probability distribution are represented as the area under a density curve. The total area under the curve is equal to [latex]1[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-5756 size-medium\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/27\/2023\/06\/22014507\/ChisqDist-1.png\" alt=\"A chi-squared distribution with df=3.\" width=\"300\" height=\"178\" \/><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-5754\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/10\/2022\/11\/03191851\/picture.png\" alt=\"The graph of a normal distribution with a mean of 0 and a standard deviation of 1.\" width=\"324\" height=\"175\" \/><\/p>\n<\/section>\n<section class=\"textbox example\">Suppose the following graph shows the distribution of heights (in inches) for women in a particular country. To find the probability that a randomly selected woman is between 60 and 66 inches tall, we would shade the area under the curve between these two values.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-1467 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5826\/2022\/10\/06200143\/Picture13.png\" alt=\"A graph of women's height in inches. 76.06% of the heights are between 60 and 66 inches.\" width=\"936\" height=\"362\" \/><\/section>\n<p>In some situations, we may use a continuous probability distribution as an approximation even when the variable is technically discrete.<\/p>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm1606\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1606&theme=lumen&iframe_resize_id=ohm1606&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm1516\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=1516&theme=lumen&iframe_resize_id=ohm1516&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<\/section>\n","protected":false},"author":8,"menu_order":10,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":3053,"module-header":"apply_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1023"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/users\/8"}],"version-history":[{"count":10,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1023\/revisions"}],"predecessor-version":[{"id":6686,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1023\/revisions\/6686"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/parts\/3053"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapters\/1023\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/media?parent=1023"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/pressbooks\/v2\/chapter-type?post=1023"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/contributor?post=1023"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/introstatstest\/wp-json\/wp\/v2\/license?post=1023"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}