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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"> Discrete Random Variables <a href="#" class="mark-page-favorite pull-right" data-pid="690" 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">Discrete 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/discrete-random-variables/main-1.jpg" style="float: left; width: 100px; height: 96px;">On this page we will explore the concept of discrete random variables and their probability distributions. We will look at how we can display probability distributions in tables or as functions. The questions are not so complicated (it is just a way of formalising the language of probability), for example, it is important not to forget that probabilities still add up to 1!</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>discrete random variables</li> <li>probability distributions</li> <li>mean (expected value)</li> <li>variance (HL only)</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/discrete-random-variables/revision-notes_discrete-random-variables.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/probability/discrete-random-variables/revision-notes_discrete-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 practises the skills from this page</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#00d239f1"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="00d239f1"> <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%;"> Discrete Random Variables <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="11-370-690" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>A unbiased coin is flipped 3 times. The outcomes are listed below</p><p>HHH , HHT , HTH , THH , HTT , THT , TTH , TTT</p><p>Complete the probability distribution below for the random variable X, where X = number of heads flipped</p></div><div class="q-answer"><table align="center" border="0" cellpadding="0" cellspacing="0" style="width:100%;"><thead><tr><th scope="col" style="text-align: center;">x</th><th scope="col" style="text-align: center;">0</th><th scope="col" style="text-align: center;">1</th><th scope="col" style="text-align: center;">2</th><th scope="col" style="text-align: center;">3</th></tr></thead><tbody><tr><td style="text-align: center;">P(X = x)</td><td style="text-align: center;"> <input type="text" style="height: auto;" data-c="0.125"> <span class="review"></span></td><td style="text-align: center;"> <input type="text" style="height: auto;" data-c="0.375"> <span class="review"></span></td><td style="text-align: center;"> <input type="text" style="height: auto;" data-c="0.375"> <span class="review"></span></td><td style="text-align: center;"> <input type="text" style="height: auto;" data-c="0.125"> <span class="review"></span></td></tr></tbody></table></div><div class="q-explanation"><p>There are 8 outcomes</p><p>TTT</p><p>P(X = 0) = 1/8</p><hr class="hidden-separator"><p>HTT , THT , TTH</p><p>P(X = 1) = 3/8</p><hr class="hidden-separator"><p>HHT , HTH , THH</p><p>P(X = 2) = 3/8</p><hr class="hidden-separator"><p>HHH</p><p>P(X = 3) = 1/8</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>A discrete random variable X has the following probability distribution</p><p>P(X = x) = <span class="math-tex">\(\frac{1}{20}x(x + 1)\)</span> , where <strong><em>x</em></strong> is 0 , 1 , 2 , 3</p><p>Complete the table below</p></div><div class="q-answer"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><thead><tr><th scope="col" style="text-align: center;">x</th><th scope="col" style="text-align: center;">0</th><th scope="col" style="text-align: center;">1</th><th scope="col" style="text-align: center;">2</th><th scope="col" style="text-align: center;">3</th></tr></thead><tbody><tr><td style="text-align: center;">P(X = x)</td><td style="text-align: center;"> <input type="text" style="height: auto;" data-c="0"> <span class="review"></span></td><td style="text-align: center;"> <input type="text" style="height: auto;" data-c="0.1"> <span class="review"></span></td><td style="text-align: center;"> <input type="text" style="height: auto;" data-c="0.3"> <span class="review"></span></td><td style="text-align: center;"> <input type="text" style="height: auto;" data-c="0.6"> <span class="review"></span></td></tr></tbody></table></div><div class="q-explanation"><p>P(X = 0) = <span class="math-tex">\(\frac{1}{20}\cdot0(0 + 1)\)</span></p><p>P(X = 1) = <span class="math-tex">\(\frac{1}{20}\cdot1(1 + 1)\)</span></p><p>P(X = 2) = <span class="math-tex">\(\frac{1}{20}\cdot2(2 + 1)\)</span></p><p>P(X = 3) = <span class="math-tex">\(\frac{1}{20}\cdot3(3 + 1)\)</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>Which is the correct function on the interval <span class="math-tex">\(x\in{0,1,2,3,4}\)</span> for the following probability distribution</p><img alt="" src="../../files/probability/discrete-random-variables/quiz/q3.png" style="width: 400px; height: 46px;"></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(\frac{1}{40} (x²+x)\)</span></label> </p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{1}{15}2^{x-1}\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\left( \begin{matrix} 4 \\ x \end{matrix} \right) 0.5^{ x }0.5^{ { 4-x } }\)</span></label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{1}{24}(2x+1)\)</span></label></p></div><div class="q-explanation"><p>If we work out P(X=1) for each of the distributions we can see which has the correct probability</p><p><span class="math-tex">\(\frac{1}{40} (1²+1)=0.05\)</span></p><p><span class="math-tex">\(\frac{1}{15}2^{1-1}\approx0.667\)</span></p><p><span class="math-tex">\(\left( \begin{matrix} 4 \\ 1 \end{matrix} \right) 0.5^{ 1 }0.5^{ { 4-1 } }=4\times0.0625=0.25\)</span></p><p><span class="math-tex">\(\frac{1}{24}(2\times1+1)=0.125\)</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 random variable <strong><em>X</em></strong> has the following probability distribution</p><p><img alt="" src="../../files/probability/discrete-random-variables/quiz/q4.png" style="width: 400px; height: 46px;"></p><p>Find the value of <strong><em>k</em></strong></p></div><div class="q-answer"><p><strong><em>k </em></strong>= <input type="text" style="height: auto;" data-c="0.35"> <span class="review"></span></p></div><div class="q-explanation"><p>All probabilities add up to 1</p><p>0.1 + 0.35 + <em><strong>k</strong></em> + 0.15 + 0.05 = 1</p><p>0.65 + <strong><em>k</em></strong> = 1</p><p><strong><em>k</em></strong> = 0.35</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 random variable <strong><em>X</em></strong> has probability function</p><p><img alt="" src="../../files/probability/discrete-random-variables/quiz/q5.png" style="width: 400px; height: 46px;"></p><p>Find P( 2 < X < 8)</p></div><div class="q-answer"><p>Find P( 2 < X < 8) = <input type="text" style="height: auto;" data-c="0.65"> <span class="review"></span></p></div><div class="q-explanation">All probabilities add up to 1, therefore</div><div class="q-explanation">q = 0.15</div><div class="q-explanation"><img alt="" src="../../files/probability/discrete-random-variables/quiz/q5a.png" style="width: 400px; height: 44px;"></div><div class="q-explanation">P( 2 < X < 8) = 0.3 + 0.2 + 0.15 = 0.65</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 function of a discrete random variable <em><strong>X</strong></em> is given by</p><p>P(X=x) = <span class="math-tex">\(\frac{x^2}{k}\)</span> for <span class="math-tex">\(x\in{1,2,3,4}\)</span></p><p>Find the value of <strong><em>k</em></strong></p></div><div class="q-answer"><p><strong><em>k</em></strong> = <input type="text" style="height: auto;" data-c="39"> <span class="review"></span></p></div><div class="q-explanation"><p>The sum of all the probabilities is 1</p><p>P(X=1) = <span class="math-tex">\(\frac{1^2}{k}\)</span></p><p>P(X=2) = <span class="math-tex">\(\frac{2^2}{k}\)</span></p><p>P(X=3) = <span class="math-tex">\(\frac{3^2}{k}\)</span></p><p>P(X=4) = <span class="math-tex">\(\frac{4^2}{k}\)</span></p><p><span class="math-tex">\(\frac{1}{k}+\frac{4}{k}+\frac{9}{k}+\frac{16}{k}=1\)</span></p><p><span class="math-tex">\(\frac{39}{k}=1\)</span></p><p>k = 39</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 random variable <em><strong>X</strong></em> has probability function</p><p><img alt="" src="../../files/probability/discrete-random-variables/quiz/q7a.png" style="width: 350px; height: 128px;"></p><p>P(X>3) = <span class="math-tex">\(\frac{a}{b}\)</span></p><p>Hint: work out the value of <strong><em>k </em></strong>first</p></div><div class="q-answer"><p>P(X>3) = <span class="math-tex">\(\frac{a}{b}\)</span></p><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="7"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="13"> <span class="review"></span></p></div><div class="q-explanation"><p>P(X = 1) = k</p><p>P(X = 2) = 2k</p><p>P(X = 3) = 3k</p><p>P(X = 4) = 3k</p><p>P(X = 5) = 4k</p><p>All probabilities add up to 1</p><p>k + 2k + 3k + 3k + 4k = 1</p><p>13k = 1</p><p>k = <span class="math-tex">\(\frac{1}{13}\)</span></p><hr class="hidden-separator"><p>P(X > 3) = P(X = 4) + P(X = 5)</p><p>P(X > 3) = 3k + 4k = <span class="math-tex">\(\frac{7}{13}\)</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 random variable <strong><em>X</em></strong> has probability function</p><p><img alt="" src="../../files/probability/discrete-random-variables/quiz/q8.png" style="width: 400px; height: 45px;"></p><p>By working out <strong><em>q</em></strong> first, find E(X)</p></div><div class="q-answer"><p>E(X) = <input type="text" style="height: auto;" data-c="3.8"> <span class="review"></span></p></div><div class="q-explanation"><p>All probabilities add up to 1</p><p><img alt="" src="../../files/probability/discrete-random-variables/quiz/q8a.png" style="width: 400px; height: 46px;"></p><p>E(X) = <span class="math-tex">\(\sum { xP(X=x) } \)</span></p><p>E(X) = 1x0.2 + 2x0.3 + 4x0.3 + 7x0.1 + 11x0.1 = 3.8</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 random variable <strong><em>X</em></strong> represents the outcome of a game.</p><p>Which of the follwoing games are <strong>FAIR</strong>?</p><p><em>Select all the possible correct answers</em></p></div><div class="q-answer"><p><label class="checkbox"><input type="checkbox"> <img alt="" src="../../files/probability/discrete-random-variables/quiz/q9a.png" style="width: 400px; height: 48px;"></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <img alt="" src="../../files/probability/discrete-random-variables/quiz/q9b.png" style="width: 400px; height: 48px;"></label></p><p><label class="checkbox"><input class="c" type="checkbox"> <img alt="" src="../../files/probability/discrete-random-variables/quiz/q9c.png" style="width: 400px; height: 44px;"></label></p><p><label class="checkbox"><input type="checkbox"> <img alt="" src="../../files/probability/discrete-random-variables/quiz/q9d.png" style="width: 400px; height: 45px;"></label></p></div><div class="q-explanation"><p>For a fair game, E(X) = 0</p><hr class="hidden-separator"><p><img alt="" src="../../files/probability/discrete-random-variables/quiz/q9a.png" style="width: 400px; height: 48px;"></p><p>E(X) = -2x0.4 + (-2)x0.05 + 0x0.05 + 1x0.1 + 2x0.3 = -0.2</p><hr class="hidden-separator"><p><img alt="" src="../../files/probability/discrete-random-variables/quiz/q9b.png" style="width: 400px; height: 48px;"></p><p>E(X) = -3x0.4 + 0x0.1 + 1x0.1 + 2x0.1 +3x0.3 = 0</p><hr class="hidden-separator"><p><img alt="" src="../../files/probability/discrete-random-variables/quiz/q9c.png" style="width: 400px; height: 44px;"></p><p>Since the distribution is symmetrical about x = 0, E(X) = 0</p><hr class="hidden-separator"><p> <img alt="" src="../../files/probability/discrete-random-variables/quiz/q9d.png" style="width: 400px; height: 45px;"></p><p>E(X) = -5x0.2 + 0x0.2 + 1x0.2 + 2x0.2 + 3x0.2 = 0.2</p><hr class="hidden-separator"></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 discrete random variable <strong><em>X</em></strong> has the following probability distribution</p><p><img alt="" src="../../files/probability/discrete-random-variables/quiz/q10.png" style="width: 400px; height: 45px;"></p><p>E(X) = 3.5</p><p>Work out <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="0.2"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="0.3"> <span class="review"></span></p></div><div class="q-explanation"><p><span class="math-tex">\(\sum { P(X=x)=1 } \)</span></p><p>0.1 + a + 0.3 + b + 0.1 = 1</p><p>a + b = 0.5</p><hr class="hidden-separator"><p>E(X) = <span class="math-tex">\(\sum { xP(X=x) } \)</span></p><p>0x0.1 + 1xa + 3x0.3 + 5xb + 9x0.1 = 3.5</p><p>a + 0.9 + 5b + 0.9 = 3.5</p><p>a + 5b = 1.7</p><hr class="hidden-separator"><p>Solve the simultaneous equations</p><p>a + b = 0.5</p><p>a + 5b = 1.7</p><p>a = 0.2</p><p>b = 0.3</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> </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 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"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL easy"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> </p> <p>The discrete random variable X has probability function</p> <p style="margin-left: 80px;"> P(X = x) = k(16 – x²) for x = 0, 1, 2, 3</p> <ol style="list-style-type:lower-alpha;"> <li>Find the value of the constant k</li> <li>Find P(1 ≤ X < 3)</li> </ol> <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"><content> </content>The sum of the probabilities = 1</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/discrete-random-variables/esq1_discretervs.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/discrete-random-variables/esq1_discretervs.pdf" width="640"></iframe></p> </section> </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"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL easy"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> </p> <p>The random variable <em>X </em>has probability function</p> <p><img alt="" src="../../files/probability/discrete-random-variables/esq2.png" style="width: 300px; height: 57px;"></p> <p>a) Find the value of <strong><em>k</em></strong></p> <p>b) Work out <span class="math-tex">\(P(X\ge 2)\)</span></p> <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>Work out some probabilities by substituting the different values of x into the function</p> <p><content>e.g. P(X=1) = 1x</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/discrete-random-variables/esq2_discretervs.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/discrete-random-variables/esq2_discretervs.pdf" width="640"></iframe></p> </section> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-border panel-has-colored-body panel-expandable panel-default"> <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> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL 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>Two boxes each contain three cards. </p> <p>The first box contains cards labelled 1,3 and 5</p> <p>The second box contains cards labelled 2, 4 and 6. </p> <p>In a game, a player draws one card at random from each box and his score, X, is the <strong>sum</strong> of the numbers on the two cards.</p> <p>a) Complete the following probability distribution</p> <p><img alt="" src="../../files/probability/discrete-random-variables/esq3.png" style="width: 600px; height: 76px;"></p> <p>b) Work out E(X)</p> <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>Work out all the possible outcomes</p> <p><content><img alt="" src="../../files/probability/discrete-random-variables/esq3expl.png" style="width: 300px; height: 189px;"></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/discrete-random-variables/esq3_discretervs.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/discrete-random-variables/esq3_discretervs.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-border panel-has-colored-body panel-expandable panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <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/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL 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>The following table shows the probability distribution of a discrete random variable <strong><em>X</em></strong></p> <p><img alt="" src="../../files/probability/discrete-random-variables/esq4.png" style="width: 600px; height: 89px;"></p> <p>If <strong><em>X </em></strong>represents the return from a game. Find <strong><em>a</em></strong> and <strong><em>b</em></strong> if the game is<strong> fair</strong>.</p> <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>We know that <span class="math-tex">\(\sum P(X)=1\)</span><content> </content></p> <p>Fair game means E(X) = 0</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/discrete-random-variables/esq4_discretervs.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/discrete-random-variables/esq4_discretervs.pdf" width="640"></iframe></p> </section> <h4> </h4> </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 5</p> </div> </div> <div class="panel-body"> <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/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"> </p> <p>The discrete random variable <em>X </em>has probability function</p> <p><span class="math-tex">\(P(X=x)=k(\frac{2}{5})^x\)</span> , for <span class="math-tex">\(x\in Z ,\quad x>0\)</span></p> <p>Work out the value of <strong><em>k</em></strong></p> <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">Work out the first few probabilities...they make an infinite geometric series<content></content></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/discrete-random-variables/esq5_discretervs.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/probability/discrete-random-variables/esq5_discretervs.pdf" width="640"></iframe></p> </section> <h4> </h4> </div> <div 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