{"id":251,"date":"2021-07-14T15:58:59","date_gmt":"2021-07-14T15:58:59","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/continuous-probability-functions\/"},"modified":"2023-12-05T09:14:39","modified_gmt":"2023-12-05T09:14:39","slug":"continuous-probability-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/continuous-probability-functions\/","title":{"raw":"What is a Continuous Probability Function?","rendered":"What is a Continuous Probability Function?"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Draw a continuous probability function for a uniform distribution<\/li>\r\n \t<li>Calculate a probability for a uniform distribution<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox examples\">\r\n<h3>Recall: Inequality Symbols<\/h3>\r\nAn inequality is a mathematical statement that compares two expressions using the ideas of greater than or less than.\r\n\r\nHere are some common inequalities seen in statistics:\r\n<ul>\r\n \t<li>&lt; indicates less than, for example x &lt; 5 indicates <em>x<\/em> is less than 5<\/li>\r\n \t<li>\u2264 indicates less than or equal to, for example <em>x<\/em> \u2264 5 indicates <em>x<\/em> is less than or equal to 5 (5 is included)<\/li>\r\n \t<li>&gt; indicates greater than, for example <em>x<\/em> &gt; 5 indicates <em>x<\/em> is greater than 5<\/li>\r\n \t<li>\u2265 indicates greater than or equal to, for example <em>x<\/em> \u2265 5 indicates <em>x<\/em> is greater than or equal to 5 (5 is included)<\/li>\r\n<\/ul>\r\n<strong>Note:<\/strong> Where you place the variable in the inequality statement can change the symbol you use.\r\n\r\nFor example:\r\n<ul>\r\n \t<li>x &lt; 5 indicates all possible numbers less than 5.<\/li>\r\n \t<li>5 &lt; <em>x<\/em> indicates that 5 is less than <em>x,<\/em> or we could rewrite this with the <em>x<\/em> on the left: <em>x<\/em> &gt; 5.<\/li>\r\n<\/ul>\r\n<strong>Note:<\/strong> how the inequality is still pointing the same direction relative to <em>x<\/em>. This statement represents all the real numbers that are greater than 5, which is easier to interpret than 5 is less than <em>x<\/em>.\r\n\r\n<\/div>\r\nWe begin by defining a continuous probability density function. We use the function notation\u00a0<em>f<\/em>(<em>x<\/em>). Intermediate algebra may have been your first formal introduction to functions. In the study of probability, the functions we study are special. We define the function <em>f<\/em>(<em>x<\/em>) so that the area between it and the <em>x<\/em>-axis is equal to a probability. Since the maximum probability is one, the maximum area is also one. <strong>For continuous probability distributions, PROBABILITY = AREA.<\/strong>\r\n<div class=\"textbox examples\">\r\n<h3>Recall: Area of a Rectangle<\/h3>\r\nArea is a measure of the surface covered by a figure. A rectangle has four sides, the figure below is an example where [latex]W[\/latex] is the width and [latex]L[\/latex] is the length.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223837\/CNX_BMath_Figure_09_04_012.png\" alt=\"A rectangle is shown. Each angle is marked with a square. The top and bottom are labeled [latex]L[\/latex], the sides are labeled [latex]W[\/latex].\" \/>\r\n\r\nThe area, [latex]A[\/latex], of a rectangle is the length times the width.\r\n<p style=\"text-align: center;\">[latex]A=L \\cdot W[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nConsider the function\u00a0[latex]f(x)\\displaystyle\\frac{{1}}{{20}}[\/latex] is a horizontal line. However, since [latex]0{\\leq}x{\\leq}20[\/latex], <em>f<\/em>(<em>x<\/em>) is restricted to the portion between [latex]x=0[\/latex] and [latex]x=20[\/latex], inclusive.\r\n\r\n<img src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/uu6y-xpe7527i#fixme#fixme#fixme\" alt=\"This shows the graph of the function f(x) = 1\/20. A horiztonal line ranges from the point (0, 1\/20) to the point (20, 1\/20). A vertical line extends from the x-axis to the end of the line at point (20, 1\/20) creating a rectangle.\" \/>\r\n\r\n[latex]f(x)=\\frac{{1}}{{20}}[\/latex] for [latex]0{\\leq}x{\\leq}20[\/latex].\r\n\r\nThe graph of\u00a0[latex]f(x)=\\frac{{1}}{{20}}[\/latex] is a horizontal line segment when [latex]0{\\leq}x{\\leq}20[\/latex].\r\n\r\nThe area between\u00a0[latex]f(x)\\frac{{1}}{{20}}[\/latex].\r\n\r\n[latex]\\displaystyle\\text{AREA}={20}{(\\frac{{1}}{{20}})}={1}[\/latex]\r\n\r\nSuppose we want to find the area between [latex]f(x)=[\/latex] and the <em>x<\/em>-axis where [latex]0&lt;x&lt;2[\/latex].\r\n\r\n<img src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/wdjw-2we7527i#fixme#fixme#fixme\" alt=\"This shows the graph of the function f(x) = 1\/20. A horiztonal line ranges from the point (0, 1\/20) to the point (20, 1\/20). A vertical line extends from the x-axis to the end of the line at point (20, 1\/20) creating a rectangle. A region is shaded inside the rectangle from x = 0 to x = 2.\" \/>\r\n\r\n[latex]\\displaystyle\\text{AREA}={({2}-{0})}{(\\frac{{1}}{{20}})}={0.1}[\/latex]\r\n\r\n[latex]\\displaystyle({2}-{0})={2}=\\text{base of a rectangle}[\/latex]\r\n\r\n<strong>Reminder:<\/strong> area of a rectangle = (base)(height).\r\n\r\nThe area corresponds to a probability. The probability that <em>x<\/em> is between zero and two is 0.1, which can be written mathematically as\u00a0[latex]P(0&lt;x&lt;2)=P(x&lt;2)=0.1[\/latex].\r\n\r\nSuppose we want to find the area between [latex]f(x)=\\frac{{1}}{{20}}[\/latex] and the x-axis where [latex]4&lt;x&lt;15[\/latex].\r\n\r\n<img src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/pgdu-j2f7527i#fixme#fixme#fixme\" alt=\"This shows the graph of the function f(x) = 1\/20. A horiztonal line ranges from the point (0, 1\/20) to the point (20, 1\/20). A vertical line extends from the x-axis to the end of the line at point (20, 1\/20) creating a rectangle. A region is shaded inside the rectangle from x = 4 to x = 15.\" \/>\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\text{AREA}={({15}-{4})}{(\\frac{{1}}{{20}})}={0.55}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\text{AREA}={({15}-{4})}{(\\frac{{1}}{{20}})}={0.55}[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]\\displaystyle{({15}-{4})}={11}=\\text{the base of a rectangle}[\/latex]<\/p>\r\nThe area corresponds to the probability [latex]P(4&lt;x&lt;15)=0.55[\/latex].\r\n\r\nSuppose we want to find [latex]P(x=15)[\/latex]. On an x-y graph, [latex]x=15[\/latex] is a vertical line. A vertical line has no width (or zero width). Therefore, [latex]P(x=15)=(\\text{base})(\\text{height})=(0){(\\frac{{1}}{{20}})}=0[\/latex]\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/8ddj-tbf7527i#fixme#fixme#fixme\" alt=\"This shows the graph of the function f(x) = 1\/20. A horiztonal line ranges from the point (0, 1\/20) to the point (20, 1\/20). A vertical line extends from the x-axis to the end of the line at point (20, 1\/20) creating a rectangle. A vertical line extends from the horizontal axis to the graph at x = 15.\" \/>\r\n\r\n[latex]P(X{\\leq}x)[\/latex] (can be written as [latex]P(X&lt;x)[\/latex] for continuous distributions) is called the cumulative distribution function or CDF. Notice the \"less than or equal to\" symbol. We can use the CDF to calculate [latex]P(X&gt;x)[\/latex]. The CDF gives \"area to the left\" and [latex]P(X&gt;x)[\/latex] gives \"area to the right.\" We calculate [latex]P(X &gt; x)[\/latex] for continuous distributions as follows: [latex]P(X&gt;x)=1\u2013P(X&lt;x)[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/8bj2-gif7527i#fixme#fixme#fixme\" alt=\"This shows the graph of the function f(x) = 1\/20. A horiztonal line ranges from the point (0, 1\/20) to the point (20, 1\/20). A vertical line extends from the x-axis to the end of the line at point (20, 1\/20) creating a rectangle. The area to the left of a value, x, is shaded.\" width=\"487\" height=\"219\" \/>\r\n\r\nLabel the graph with\u00a0<em>f<\/em>(<em>x<\/em>) and <em>x<\/em>. Scale the <em>x<\/em> and <em>y<\/em> axes with the maximum <em>x<\/em> and <em>y<\/em> values.<em>f<\/em>(<em>x<\/em>) =\u00a0[latex]\\displaystyle\\frac{{1}}{{20}}[\/latex], [latex]0{\\leq}x{\\leq}20[\/latex].\r\n\r\nTo calculate the probability that\u00a0<em>x<\/em> is between two values, look at the following graph. Shade the region between [latex]x=2.3[\/latex] and [latex]x=12.7[\/latex]. Then calculate the shaded area of a rectangle.\r\n\r\n<img src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/pft7-9mf7527i#fixme#fixme#fixme\" alt=\"This shows the graph of the function f(x) = 1\/20. A horiztonal line ranges from the point (0, 1\/20) to the point (20, 1\/20). A vertical line extends from the x-axis to the end of the line at point (20, 1\/20) creating a rectangle. A region is shaded inside the rectangle from x = 2.3 to x = 12.7\" \/>\r\n\r\n[latex]\\displaystyle{P}{({2.3}{&lt;}{x}{&lt;}{12.7})}={(\\text{base})}{(\\text{height})}={({12.7}-{2.3})}{(\\frac{{1}}{{20}})}={0.52}[\/latex]\r\n\r\n<\/div>\r\nThis video will help you summarize what you just read.\r\n\r\n<iframe src=\"\/\/plugin.3playmedia.com\/show?mf=7115016&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=j8XLYFzTJzE&amp;video_target=tpm-plugin-p2bf90ab-j8XLYFzTJzE\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nConsider the function\u00a0[latex]f(x)\\frac{{1}}{{8}}[\/latex] for [latex]0{\\leq}x{\\leq}8[\/latex]. Draw the graph of <em>f<\/em>(<em>x<\/em>) and find [latex]P(2.5&lt;x&lt;7.5)[\/latex].\r\n\r\n[reveal-answer q=\"287031\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"287031\"]\r\n\r\n<img src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/zf3u-mqf7527i#fixme#fixme#fixme\" alt=\"\" \/>\r\n\r\n[latex]P (2.5&lt;x&lt;7.5)=0.625[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Draw a continuous probability function for a uniform distribution<\/li>\n<li>Calculate a probability for a uniform distribution<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox examples\">\n<h3>Recall: Inequality Symbols<\/h3>\n<p>An inequality is a mathematical statement that compares two expressions using the ideas of greater than or less than.<\/p>\n<p>Here are some common inequalities seen in statistics:<\/p>\n<ul>\n<li>&lt; indicates less than, for example x &lt; 5 indicates <em>x<\/em> is less than 5<\/li>\n<li>\u2264 indicates less than or equal to, for example <em>x<\/em> \u2264 5 indicates <em>x<\/em> is less than or equal to 5 (5 is included)<\/li>\n<li>&gt; indicates greater than, for example <em>x<\/em> &gt; 5 indicates <em>x<\/em> is greater than 5<\/li>\n<li>\u2265 indicates greater than or equal to, for example <em>x<\/em> \u2265 5 indicates <em>x<\/em> is greater than or equal to 5 (5 is included)<\/li>\n<\/ul>\n<p><strong>Note:<\/strong> Where you place the variable in the inequality statement can change the symbol you use.<\/p>\n<p>For example:<\/p>\n<ul>\n<li>x &lt; 5 indicates all possible numbers less than 5.<\/li>\n<li>5 &lt; <em>x<\/em> indicates that 5 is less than <em>x,<\/em> or we could rewrite this with the <em>x<\/em> on the left: <em>x<\/em> &gt; 5.<\/li>\n<\/ul>\n<p><strong>Note:<\/strong> how the inequality is still pointing the same direction relative to <em>x<\/em>. This statement represents all the real numbers that are greater than 5, which is easier to interpret than 5 is less than <em>x<\/em>.<\/p>\n<\/div>\n<p>We begin by defining a continuous probability density function. We use the function notation\u00a0<em>f<\/em>(<em>x<\/em>). Intermediate algebra may have been your first formal introduction to functions. In the study of probability, the functions we study are special. We define the function <em>f<\/em>(<em>x<\/em>) so that the area between it and the <em>x<\/em>-axis is equal to a probability. Since the maximum probability is one, the maximum area is also one. <strong>For continuous probability distributions, PROBABILITY = AREA.<\/strong><\/p>\n<div class=\"textbox examples\">\n<h3>Recall: Area of a Rectangle<\/h3>\n<p>Area is a measure of the surface covered by a figure. A rectangle has four sides, the figure below is an example where [latex]W[\/latex] is the width and [latex]L[\/latex] is the length.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223837\/CNX_BMath_Figure_09_04_012.png\" alt=\"A rectangle is shown. Each angle is marked with a square. The top and bottom are labeled [latex]L[\/latex], the sides are labeled [latex]W[\/latex].\" \/><\/p>\n<p>The area, [latex]A[\/latex], of a rectangle is the length times the width.<\/p>\n<p style=\"text-align: center;\">[latex]A=L \\cdot W[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Consider the function\u00a0[latex]f(x)\\displaystyle\\frac{{1}}{{20}}[\/latex] is a horizontal line. However, since [latex]0{\\leq}x{\\leq}20[\/latex], <em>f<\/em>(<em>x<\/em>) is restricted to the portion between [latex]x=0[\/latex] and [latex]x=20[\/latex], inclusive.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/uu6y-xpe7527i#fixme#fixme#fixme\" alt=\"This shows the graph of the function f(x) = 1\/20. A horiztonal line ranges from the point (0, 1\/20) to the point (20, 1\/20). A vertical line extends from the x-axis to the end of the line at point (20, 1\/20) creating a rectangle.\" \/><\/p>\n<p>[latex]f(x)=\\frac{{1}}{{20}}[\/latex] for [latex]0{\\leq}x{\\leq}20[\/latex].<\/p>\n<p>The graph of\u00a0[latex]f(x)=\\frac{{1}}{{20}}[\/latex] is a horizontal line segment when [latex]0{\\leq}x{\\leq}20[\/latex].<\/p>\n<p>The area between\u00a0[latex]f(x)\\frac{{1}}{{20}}[\/latex].<\/p>\n<p>[latex]\\displaystyle\\text{AREA}={20}{(\\frac{{1}}{{20}})}={1}[\/latex]<\/p>\n<p>Suppose we want to find the area between [latex]f(x)=[\/latex] and the <em>x<\/em>-axis where [latex]0<x<2[\/latex].\n\n<img decoding=\"async\" src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/wdjw-2we7527i#fixme#fixme#fixme\" alt=\"This shows the graph of the function f(x) = 1\/20. A horiztonal line ranges from the point (0, 1\/20) to the point (20, 1\/20). A vertical line extends from the x-axis to the end of the line at point (20, 1\/20) creating a rectangle. A region is shaded inside the rectangle from x = 0 to x = 2.\" \/><\/p>\n<p>[latex]\\displaystyle\\text{AREA}={({2}-{0})}{(\\frac{{1}}{{20}})}={0.1}[\/latex]<\/p>\n<p>[latex]\\displaystyle({2}-{0})={2}=\\text{base of a rectangle}[\/latex]<\/p>\n<p><strong>Reminder:<\/strong> area of a rectangle = (base)(height).<\/p>\n<p>The area corresponds to a probability. The probability that <em>x<\/em> is between zero and two is 0.1, which can be written mathematically as\u00a0[latex]P(0<x<2)=P(x<2)=0.1[\/latex].\n\nSuppose we want to find the area between [latex]f(x)=\\frac{{1}}{{20}}[\/latex] and the x-axis where [latex]4<x<15[\/latex].\n\n<img decoding=\"async\" src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/pgdu-j2f7527i#fixme#fixme#fixme\" alt=\"This shows the graph of the function f(x) = 1\/20. A horiztonal line ranges from the point (0, 1\/20) to the point (20, 1\/20). A vertical line extends from the x-axis to the end of the line at point (20, 1\/20) creating a rectangle. A region is shaded inside the rectangle from x = 4 to x = 15.\" \/><\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\text{AREA}={({15}-{4})}{(\\frac{{1}}{{20}})}={0.55}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\text{AREA}={({15}-{4})}{(\\frac{{1}}{{20}})}={0.55}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle{({15}-{4})}={11}=\\text{the base of a rectangle}[\/latex]<\/p>\n<p>The area corresponds to the probability [latex]P(4<x<15)=0.55[\/latex].\n\nSuppose we want to find [latex]P(x=15)[\/latex]. On an x-y graph, [latex]x=15[\/latex] is a vertical line. A vertical line has no width (or zero width). Therefore, [latex]P(x=15)=(\\text{base})(\\text{height})=(0){(\\frac{{1}}{{20}})}=0[\/latex]\n\n<img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/8ddj-tbf7527i#fixme#fixme#fixme\" alt=\"This shows the graph of the function f(x) = 1\/20. A horiztonal line ranges from the point (0, 1\/20) to the point (20, 1\/20). A vertical line extends from the x-axis to the end of the line at point (20, 1\/20) creating a rectangle. A vertical line extends from the horizontal axis to the graph at x = 15.\" \/><\/p>\n<p>[latex]P(X{\\leq}x)[\/latex] (can be written as [latex]P(X<x)[\/latex] for continuous distributions) is called the cumulative distribution function or CDF. Notice the &#8220;less than or equal to&#8221; symbol. We can use the CDF to calculate [latex]P(X>x)[\/latex]. The CDF gives &#8220;area to the left&#8221; and [latex]P(X>x)[\/latex] gives &#8220;area to the right.&#8221; We calculate [latex]P(X > x)[\/latex] for continuous distributions as follows: [latex]P(X>x)=1\u2013P(X<x)[\/latex].\n\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/8bj2-gif7527i#fixme#fixme#fixme\" alt=\"This shows the graph of the function f(x) = 1\/20. A horiztonal line ranges from the point (0, 1\/20) to the point (20, 1\/20). A vertical line extends from the x-axis to the end of the line at point (20, 1\/20) creating a rectangle. The area to the left of a value, x, is shaded.\" width=\"487\" height=\"219\" \/><\/p>\n<p>Label the graph with\u00a0<em>f<\/em>(<em>x<\/em>) and <em>x<\/em>. Scale the <em>x<\/em> and <em>y<\/em> axes with the maximum <em>x<\/em> and <em>y<\/em> values.<em>f<\/em>(<em>x<\/em>) =\u00a0[latex]\\displaystyle\\frac{{1}}{{20}}[\/latex], [latex]0{\\leq}x{\\leq}20[\/latex].<\/p>\n<p>To calculate the probability that\u00a0<em>x<\/em> is between two values, look at the following graph. Shade the region between [latex]x=2.3[\/latex] and [latex]x=12.7[\/latex]. Then calculate the shaded area of a rectangle.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/pft7-9mf7527i#fixme#fixme#fixme\" alt=\"This shows the graph of the function f(x) = 1\/20. A horiztonal line ranges from the point (0, 1\/20) to the point (20, 1\/20). A vertical line extends from the x-axis to the end of the line at point (20, 1\/20) creating a rectangle. A region is shaded inside the rectangle from x = 2.3 to x = 12.7\" \/><\/p>\n<p>[latex]\\displaystyle{P}{({2.3}{<}{x}{<}{12.7})}={(\\text{base})}{(\\text{height})}={({12.7}-{2.3})}{(\\frac{{1}}{{20}})}={0.52}[\/latex]\n\n<\/div>\n<p>This video will help you summarize what you just read.<\/p>\n<p><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=7115016&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=j8XLYFzTJzE&amp;video_target=tpm-plugin-p2bf90ab-j8XLYFzTJzE\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Consider the function\u00a0[latex]f(x)\\frac{{1}}{{8}}[\/latex] for [latex]0{\\leq}x{\\leq}8[\/latex]. Draw the graph of <em>f<\/em>(<em>x<\/em>) and find [latex]P(2.5<x<7.5)[\/latex].\n\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q287031\">Show Solution<\/span><\/p>\n<div id=\"q287031\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/zf3u-mqf7527i#fixme#fixme#fixme\" alt=\"\" \/><\/p>\n<p>[latex]P (2.5<x<7.5)=0.625[\/latex]\n\n<\/div>\n<\/div>\n<\/div>\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-251\">\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>OpenStax, Statistics, Continuous Probability Functions. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/statistics\/pages\/5-1-continuous-probability-functions\">https:\/\/openstax.org\/books\/statistics\/pages\/5-1-continuous-probability-functions<\/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\/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><li>College Algebra. <strong>Authored by<\/strong>: Jay Abramson, et al. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-describe-solutions-to-inequalities-2\/\">https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-describe-solutions-to-inequalities-2\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/prealgebra\/pages\/1-introduction\">https:\/\/openstax.org\/books\/prealgebra\/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\/prealgebra\/pages\/1-introduction<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">All rights reserved content<\/div><ul class=\"citation-list\"><li>Continuous Probability distribution intro. <strong>Authored by<\/strong>: Khan Academy. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.youtube.com\/watch?v=j8XLYFzTJzE\">https:\/\/www.youtube.com\/watch?v=j8XLYFzTJzE<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/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":8,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"OpenStax, Statistics, Continuous Probability Functions\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/statistics\/pages\/5-1-continuous-probability-functions\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/statistics\/pages\/1-introduction\"},{\"type\":\"copyrighted_video\",\"description\":\"Continuous Probability distribution intro\",\"author\":\"Khan Academy\",\"organization\":\"\",\"url\":\"https:\/\/www.youtube.com\/watch?v=j8XLYFzTJzE\",\"project\":\"\",\"license\":\"pd\",\"license_terms\":\"\"},{\"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\"},{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/prealgebra\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/prealgebra\/pages\/1-introduction\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Jay Abramson, et al\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-describe-solutions-to-inequalities-2\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-251","chapter","type-chapter","status-publish","hentry"],"part":249,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/251","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\/251\/revisions"}],"predecessor-version":[{"id":3609,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/251\/revisions\/3609"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/249"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/251\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=251"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=251"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=251"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=251"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}