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Distribution</a></label></li></ul></div> <div class="hidden-xs hidden-sm"> <button class="btn btn-default btn-block text-xs-center" data-toggle="modal" data-target="#modal-feedback" style="margin-bottom: 10px"><i class="fa fa-send"></i> Feedback</button> </div> </div> <div class="col-md-9" id="main-column"> <h1 class="page_title"> Continuous Random Variables <a href="#" class="mark-page-favorite pull-right" data-pid="691" title="Mark as favorite" onclick="return false;"><i class="fa fa-star-o"></i></a> </h1> <ol class="breadcrumb"> <li><a href="../../../mathsanalysis.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../549/stats-probability.html">Stats & Probability</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Continuous Random Variables</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <p><img alt="" src="../../files/probability/continuousrvs/main.png" style="float: left; width: 100px; height: 98px;"> A random variable is continuous if the values can be expressed as an interval. Calculations are for continuous variables, for example heights or times, that is, something that can be measured rather than counted. In order to calculate probabilities, we need to find the area under graphs, so it is important that you are confident with integration techniques before working through this page. In the exam, questions are usually the long part B questions. Typical questions are given in the exam-style questions below.</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <div> <p>On this page, you should learn about</p> <ul> <li>continuous random variables and their probability density functions</li> <li>mode of continuous random variables</li> <li>median of continuous random variables</li> <li>mean of continuous random variables</li> <li>variance of continuous random variables</li> </ul> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/probability/continuousrvs/revision-notes_continuous_random_variables.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/probability/continuousrvs/revision-notes_continuous_random_variables.pdf" target="_blank">here</a></p> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <p>Here is a quiz that gets you to check the initial conditions for continuous random variables</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#e156f98d"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="e156f98d"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Continuous Random Variables 1 <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-542-691" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>The graph below shows a continuous random variable, X</p><p>Find the value of <strong><em>a</em></strong></p><p><img alt="" src="../../files/probability/continuousrvs/quiz1/esq1" style="width: 300px; height: 302px;"></p></div><div class="q-answer"><p><em><strong>a</strong></em> = <input type="text" style="height: auto;" data-c="0.2"> <span class="review"></span></p></div><div class="q-explanation"><p>The area under the graph equals 1</p><p>5a = 1</p><p>a = <span class="math-tex">\(\frac{1}{5}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph below shows a continuous random variable, X</p><p>Find the value of <strong><em>a</em></strong></p><p><strong><em><img alt="" src="../../files/probability/continuousrvs/quiz1/esq2" style="width: 300px; height: 436px;"></em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="0.5"> <span class="review"></span></p></div><div class="q-explanation"><p>The area under the graph equals 1</p><p><span class="math-tex">\(\frac{4a}{2}=1\)</span></p><p>a = <span class="math-tex">\(\frac{1}{2}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph below shows a continuous random variable, X, where the probability density function is given by</p><p style="margin-left: 80px;"><span class="math-tex"><a\\otherwise\\\end{matrix}\)< span=""></a\\otherwise\\\end{matrix}\)<></span><span class="math-tex"><span class="math-tex">\(f\left(x\right)=\left\{\begin{matrix}\frac{x^2}{9}\\0\\\end{matrix}\ \ \ \ \ \ \right.\begin{matrix}0\le x<a\\otherwise\\\end{matrix}\)</span><a\\otherwise\\\end{matrix}\)< span=""></a\\otherwise\\\end{matrix}\)<></span></p><p>Find the value of <strong><em>a</em></strong></p><p><strong><em><img alt="" src="../../files/probability/continuousrvs/quiz1/esq3" style="width: 300px; height: 372px;"></em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>The area under the graph equals 1</p><p><span class="math-tex">\(\int_{0}^{a}{\frac{x^2}{9}dx=1}\)</span></p><p><span class="math-tex">\(\left[\frac{x^3}{27}\right]_0^a=1\)</span></p><p><span class="math-tex">\(\frac{a^3}{27}=1\)</span></p><p><span class="math-tex">\(a^3=27\)</span></p><p>a = 3</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph below shows a continuous random variable, X</p><p>Find the value of <strong><em>a</em></strong></p><p><strong><em><img alt="" src="../../files/probability/continuousrvs/quiz1/esq4" style="width: 300px; height: 177px;"></em></strong></p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(\frac{\sqrt{6}}{3}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{\sqrt{2}}{2}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span>1</span></label> </p><p><label class="radio"> <input type="radio"> <span><span class="math-tex">\(\frac{1}{2}\)</span></span></label> </p></div><div class="q-explanation"><p>The area under the graph equals 1</p><p><img alt="" src="../../files/probability/continuousrvs/quiz1/esq4a" style="width: 300px; height: 193px;"></p><p><span class="math-tex">\(a^2+\frac{a^2}{2}=1\)</span></p><p><span class="math-tex">\(\frac{3a^2}{2}=1\)</span></p><p><span class="math-tex">\(a^2=\frac{2}{3}\)</span></p><p><span class="math-tex">\(a=\frac{\sqrt{2}}{\sqrt{3}}\)</span></p><p>Rationalise the denominator</p><p><span class="math-tex">\(a=\frac{\sqrt{2}\sqrt{3}}{3}\)</span></p><p><span class="math-tex">\(a=\frac{\sqrt{6}}{3}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The probability density function of X is given by</p><p style="margin-left: 80px;"><span class="math-tex"><a\\otherwise\\\end{matrix}\)< span=""></a\\otherwise\\\end{matrix}\)<></span><span class="math-tex"><span class="math-tex">\(f\left(x\right)=\left\{\begin{matrix}0.5\\0\\\end{matrix}\ \ \ \ \ \ \right.\begin{matrix}0\le x<a\\otherwise\\\end{matrix}\)</span><a\\otherwise\\\end{matrix}\)< span=""></a\\otherwise\\\end{matrix}\)<></span></p><p>Find the value of <strong><em>a </em></strong> that makes this a continuous random variable</p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p>The area under the graph of the function equals 1</p><p>0.5 x 2 = 1</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The probability density function of X is given by</p><p style="margin-left: 80px;"><span class="math-tex"><a\\otherwise\\\end{matrix}\)< span=""></a\\otherwise\\\end{matrix}\)<></span><span class="math-tex"><span class="math-tex">\(f\left(x\right)=\left\{\begin{matrix}\frac{x}{8}\\0\\\end{matrix}\ \ \ \ \ \ \right.\begin{matrix}0\le x<a\\otherwise\\\end{matrix}\)</span><a\\otherwise\\\end{matrix}\)< span=""></a\\otherwise\\\end{matrix}\)<></span></p><p>Find the value of <strong><em>a </em></strong> that makes this a continuous random variable</p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p>The area under the graph of the function equals 1</p><p>There is no need to use integration. We can find the area of the triangle</p><p><span class="math-tex">\(\frac{a^2}{16}=1\)</span></p><p>a² = 16</p><p>a = 4</p><p><img alt="" src="../../files/probability/continuousrvs/quiz1/esq6" style="width: 300px; height: 293px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The probability density function of X is given by</p><p style="margin-left: 80px;"><span class="math-tex">\(f\left(x\right)=\left\{\begin{matrix}e^x\\0\\\end{matrix}\ \ \ \ \ \ \right.\begin{matrix}0\le x<lna\\otherwise\\\end{matrix}\)</span></p><p style="margin-left: 80px;"><span class="math-tex"><ln\ a=""></ln\></span></p><p>Find the value of <strong><em>a </em></strong> that makes this a continuous random variable</p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p>The area under the graph of the function equals 1</p><p><span class="math-tex">\(\int_{0}^{lna}{e^xdx=1}\)</span></p><p><span class="math-tex">\(\left[e^x\right]_0^{lna}=1\)</span></p><p><span class="math-tex">\(e^{lna}-e^0=1\)</span></p><p>a - 1 = 1</p><p>a = 2</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The probability density function of X is given by</p><p style="margin-left: 80px;"><span class="math-tex">\(f\left(x\right)=\left\{\begin{matrix}\frac{1}{6}x\\0\\\end{matrix}\ \ \ \ \ \ \right.\begin{matrix}a\le x<4\\otherwise\\\end{matrix}\)</span></p><p>Find the value of <strong><em>a </em></strong> that makes this a continuous random variable</p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p>The area under the graph of the function equals 1</p><p>We do not need to use integration to solve this</p><p><img alt="" src="../../files/probability/continuousrvs/quiz1/esq8" style="width: 300px; height: 311px;"></p><p><span class="math-tex">\(\frac{4\times\frac{4}{6}}{2}-\frac{a\times\frac{a}{6}}{2}=1\)</span></p><p><span class="math-tex">\(\frac{16-a^2}{12}=1\)</span></p><p>16 - a² = 12</p><p>a² = 4</p><p>a = 2</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph below shows a continuous random variable, X</p><p>Find <span class="math-tex">\(P(1\le X<1.5)\)</span></p><p><img alt="" src="../../files/probability/continuousrvs/quiz1/esq9" style="width: 300px; height: 336px;"></p></div><div class="q-answer"><p>probability = <input type="text" style="height: auto;" data-c="0.25"> <span class="review"></span></p></div><div class="q-explanation"><p>We find the area under the graph between 1 and 1.5</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph below shows a continuous random variable, X</p><p>Find <span class="math-tex">\(P(0.5\le X<1.5)\)</span></p><p><img alt="" src="../../files/probability/continuousrvs/quiz1/esq10" style="width: 300px; height: 451px;"></p></div><div class="q-answer"><p>probability = <input type="text" style="height: auto;" data-c="0.5"> <span class="review"></span></p></div><div class="q-explanation"><p>We find the area under the graph between 0.5 and 1.5</p><p>Area <span class="math-tex">\(\frac{1.5\times 0.75}{2}-\frac{0.5\times 0.25}{2}=0.5\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator"> <p>Here is a quiz about finding the mean, mode and median from a continuous random variable</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#589bc593"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="589bc593"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Continuous Random Variables 2 - mean, mode and median <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-541-691" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>The graph below shows the continuous random variable X.</p><p>Find E(X)</p><p><img alt="" src="../../files/probability/continuousrvs/quiz2/esq1" style="width: 400px; height: 368px;"></p></div><div class="q-answer"><p>E(X) = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p>E(X) is the mean.</p><p>Since this is a symmetrical distribution, there is no need to use any integration. It is the mid-value.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph below shows the continuous random variable X.</p><p>Find E(X)</p><p><img alt="" src="../../files/probability/continuousrvs/quiz2/esq2" style="width: 400px; height: 355px;"></p></div><div class="q-answer"><p>E(X) = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p>E(X) is the mean.</p><p>Since this is a symmetrical distribution, there is no need to use any integration. It is the mid-value.</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph below shows the continuous random variable X.</p><p>Find the median.</p><p><img alt="" src="../../files/probability/continuousrvs/quiz2/esq1" style="width: 400px; height: 368px;"></p></div><div class="q-answer"><p>Median = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p>It is the middle value</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph below shows the continuous random variable X.</p><p>Find the mode</p><p><img alt="" src="../../files/probability/continuousrvs/quiz2/esq2" style="width: 400px; height: 355px;"></p></div><div class="q-answer"><p>Mode = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p>Mode is the value at which the probability density function reaches a maximum</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph below shows the continuous random variable X.</p><p>Find the mode</p><p><img alt="" src="../../files/probability/continuousrvs/quiz2/esq5" style="width: 400px; height: 406px;"></p></div><div class="q-answer"><p>Mode = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>Mode is the value at which the probability density function reaches a maximum</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph below shows the continuous random variable X.</p><p>Find the median</p><p><img alt="" src="../../files/probability/continuousrvs/quiz2/esq2" style="width: 400px; height: 355px;"></p></div><div class="q-answer"><p>Median = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p></div><div class="q-explanation"><p>It is the middle value</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph below shows the continuous random variable X.</p><p>Find the mode</p><p><img alt="" src="../../files/probability/continuousrvs/quiz2/esq7" style="width: 400px; height: 429px;"></p></div><div class="q-answer"><p>Mode = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p>Mode is the value at which the probability density function reaches a maximum</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph below shows the continuous random variable X.</p><p>Find the median</p><p><img alt="" src="../../files/probability/continuousrvs/quiz2/esq7" style="width: 400px; height: 429px;"></p></div><div class="q-answer"><p><label class="radio"> <input class="c" type="radio"> <span><span class="math-tex">\(\sqrt{2}\)</span></span></label> </p><p><label class="radio"> <input type="radio"> <span>1</span></label> </p><p><label class="radio"> <input type="radio"> <span>1.5</span></label> </p><p><label class="radio"> <input type="radio"> <span>2</span></label> </p></div><div class="q-explanation"><p>Let m = median</p><p><span class="math-tex">\(\frac{m\times \frac{m}{2}}{2}=\frac{1}{2}\)</span></p><p>m² = 2</p><p><span class="math-tex">\(m=\sqrt{2}\)</span></p><p><img alt="" src="../../files/probability/continuousrvs/quiz2/esq8" style="width: 400px; height: 493px;"></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph below shows the continuous random variable X where</p><p style="margin-left: 80px;"><span class="math-tex">\(f\left(x\right)=\left\{\begin{matrix}\frac{x}{2}\\0\\\end{matrix}\ \ \ \ \ \ \ \right.\begin{matrix}0\le x<2\\otherwise\\\end{matrix}\)</span></p><p>E(X) = <span class="math-tex">\(\frac{a}{b}\)</span> where <span class="math-tex">\(\frac{a}{b}\)</span> is a fraction in its simplest terms.</p><p>Find <strong><em>a</em></strong> and <strong><em>b</em></strong></p><p><img alt="" src="../../files/probability/continuousrvs/quiz2/esq7" style="width: 400px; height: 429px;"></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(E\left(X\right)=\int_{0}^{2}{\frac{x}{2}\bullet x d x}\)</span></p><p><span class="math-tex">\(E\left(X\right)=\int_{0}^{2}{\frac{x^2}{2} d x}\)</span></p><p><span class="math-tex">\(E\left(X\right)=\left[\frac{x^3}{6}\right]_0^2\)</span></p><p><span class="math-tex">\(E(X)=\frac{8}{6}=\frac{4}{3}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The continuous random variable X is given by the following probability density function</p><p style="margin-left: 80px;"><span class="math-tex">\(f\left(x\right)=\left\{\begin{matrix}1-x\\0\\\end{matrix}\ \ \ \ \ \ \ \right.\begin{matrix}0\le x<1\\otherwise\\\end{matrix}\)</span></p><p>E(X) = <span class="math-tex">\(\frac{a}{b}\)</span> where <span class="math-tex">\(\frac{a}{b}\)</span> is a fraction in its simplest terms.</p><p>Find <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(E\left(X\right)=\int_{0}^{1}{(1-x) x d x}\)</span></p><p><span class="math-tex">\(E\left(X\right)=\int_{0}^{1}{(x-x^2) d x}\)</span></p><p><span class="math-tex">\(E\left(X\right)=\left[\frac{x^2}{2}-\frac{x^3}{3}\right]_0^1\)</span></p><p><span class="math-tex">\(E(X)=\frac{1}{2}-\frac{1}{3}=\frac{1}{6}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam-style Questions</p> </div> </div> <div class="panel-body"> <div> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="887"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p><span calibri="" style="font-size:11.0pt;line-height:115%; font-family:">The probability density function of X is given by </span></p> <p><span class="math-tex">\(f\left(x\right)=\left\{\begin{matrix}ax^n\\0\\\end{matrix}\ \ \ \ \ \ \ \right.\begin{matrix}0\le x<1\\otherwise\\\end{matrix}\)</span></p> <p>a) Show that <em><strong>a = n + 1</strong></em></p> <p>b) Given that E(X) = 0.75, find <em><strong>a</strong></em> and <em><strong>n</strong></em></p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span calibri="" style="font-size:11.0pt;line-height:115%; font-family:">For continuous random variables, the total area under the function = 1</span></p> <p><content><span class="math-tex">\(\int_{-\infty}^{\infty}{f\left(x\right)dx=1}\)</span> </content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/continuousrvs/esq_continuous_rvs3.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/continuousrvs/esq_continuous_rvs3.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="885"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p><span calibri="" style="font-size:11.0pt;line-height:115%; font-family:">The continuous random variable X has a probability density function given by</span></p> <p><span class="math-tex">\(f\left(x\right)=\left\{\begin{matrix}a\\asin\left(\frac{\pi x}{4}\right)\\0\\\end{matrix}\ \ \ \ \ \ \ \right.\begin{matrix}0\le x<2\\2\le x<4\\otherwise\\\end{matrix}\)</span></p> <p>a) <span calibri="" style="font-size:11.0pt;line-height:115%; font-family:">Draw a sketch of </span><strong><em>f</em></strong></p> <p>b) <span calibri="" style="font-size:11.0pt;line-height:115%; font-family:">Show that the value of </span> <span class="math-tex">\(a=\frac{\pi}{2\pi+4}\)</span></p> <p>c) Find E(X)</p> <p>d) <span calibri="" style="font-size:11.0pt;line-height:115%; font-family:">Find the exact value of the median of X</span></p> <p>e) Find <span class="math-tex">\(P(X\le2|X\le3)\)</span></p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>b) The area under the graph of <strong><em>f</em></strong> equals 1 (sum of probabilities=1). Use this fact to find <strong><em>a</em></strong></p> <p><content>c) <span class="math-tex">\(E\left(X\right)=\int_{-\infty}^{\infty}xf\left(x\right)dx\)</span></content></p> <p>d) <span class="math-tex">\(\int_{-\infty}^{m}{f\left(x\right)dx=0.5}\)</span></p> <p>From the graph it is obvious that the median occurs between <span class="math-tex">\(0 \le x<2\)</span>, so there is no need to use Calculus</p> <p><img alt="" src="../../files/probability/continuousrvs/esq1c" style="width: 400px; height: 383px;"></p> <p>e) <span class="math-tex">\(P(X\le2|X\le3)=\frac{P(X\le2)}{P(X\le3)}\)</span></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/continuousrvs/esq_continuous_rvs1.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/continuousrvs/esq_continuous_rvs1.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="886"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="Calculator"></p> <p><span calibri="" style="font-size:11.0pt;line-height:115%; font-family:">The continuous random variable X has a probability density function given by</span></p> <p><span class="math-tex">\(f\left(x\right)=\left\{\begin{matrix}k\bullet a r c o s\left(x\right)\\0\\\end{matrix}\ \ \ \ \ \ \ \right.\begin{matrix}-1\le x<1\\otherwise\\\end{matrix}\)</span></p> <p>a) Draw a sketch of <strong><em>f</em></strong></p> <p>b) State the mode of <strong><em>X</em></strong></p> <p>c) Find <span class="math-tex">\(\int a r c o s\left(x\right)dx\)</span></p> <p>d) Find E(X)</p> <p>e) Find Var(X)</p> <hr class="hidden-separator"> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>c) You will need to use Integration by parts and make the 'other part' 1</p> <p><span class="math-tex">\(\int{1\bullet a r c o s\left(x\right)dx}\)</span><content></content></p> <p>d) and e) You can use your graphical calculator to work these out.</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/probability/continuousrvs/esq_continuous_rvs2.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/continuousrvs/esq_continuous_rvs2.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> </div> </div> <div class="page-container panel-self-assessment" data-id="691"> <div class="panel-heading">MY PROGRESS</div> <div class="panel-body understanding-rate"> <div class="msg"></div> <label class="label-lg">Self-assessment</label><p>How much of <strong>Continuous Random Variables</strong> have you understood?</p><div 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