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</div><h2>HL Paper 3</h2><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The weight of tea in <em>Supermug</em> tea bags has a normal distribution with mean 4.2 g and standard deviation 0.15 g. The weight of tea in <em>Megamug</em> tea bags has a normal distribution with mean 5.6 g and standard deviation 0.17 g.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that a randomly chosen <em>Supermug</em> tea bag contains more than 3.9 g of tea.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that, of two randomly chosen <em>Megamug</em> tea bags, one contains more than 5.4 g of tea and one contains less than 5.4 g of tea.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that five randomly chosen <em>Supermug</em> tea bags contain a total of less than 20.5 g of tea.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that the total weight of tea in seven randomly chosen <em>Supermug</em> tea bags is more than the total weight in five randomly chosen <em>Megamug</em> tea bags.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Two species of plant, \(A\) and \(B\), are identical in appearance though it is known that the mean length of leaves from a plant of species \(A\) is \(5.2\) cm, whereas the mean length of leaves from a plant of species \(B\) is \(4.6\) cm. Both lengths can be modelled by normal distributions with standard deviation \(1.2\) cm.</p>
<p>In order to test whether a particular plant is from species \(A\) or species \(B\), \(16\) leaves are collected at random from the plant. The length, \(x\), of each leaf is measured and the mean length evaluated. A one-tailed test of the sample mean, \(\bar X\), is then performed at the \(5\% \) level, with the hypotheses: \({H_0}:\mu = 5.2\) and \({H_1}:\mu < 5.2\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the critical region for this test.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">It is now known that in the area in which the plant was found \(90\% \) of all the plants are of species \(A\) and \(10\% \) are of species \(B\).</p>
<p class="p1">Find the probability that \(\bar X\) will fall within the critical region of the test.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">If, having done the test, the sample mean is found to lie within the critical region, find the probability that the leaves came from a plant of species \(A\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable \(X \sim {\text{Po}}(m)\). Given that P(<em>X </em>= <em>k </em>−1) = P(<em>X </em>= <em>k </em>+1), where <em>k </em>is a positive integer,</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">show that \({m^2} = k(k + 1)\);</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">hence show that the mode of <em>X </em>is <em>k </em>.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">A traffic radar records the speed, \(v\) kilometres per hour (\({\text{km}}\,{{\text{h}}^{-{\text{1}}}}\)), of cars on a section of a road.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The following table shows a summary of the results for a random sample of 1000 cars whose speeds were recorded on a given day.</span></p>
<p style="font: normal normal normal 20.5px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/Schermafbeelding_2014-09-18_om_07.17.39.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Using the data in the table,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) show that an estimate of the mean speed of the sample is 113.21 \({\text{km}}\,{{\text{h}}^{-{\text{1}}}}\);</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) find an estimate of the variance of the speed of the cars on this section of the road.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the 95% confidence interval, \(I\), for the mean speed.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Let \(J\) be the 90% confidence interval for the mean speed.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Without calculating \(J\), explain why \(J \subset I\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>X</em> has the distribution \({\text{B}}(n{\text{ , }}p)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) Show that \(\frac{{{\text{P}}(X = x)}}{{{\text{P}}(X = x - 1)}} = \frac{{(n - x + 1)p}}{{x(1 - p)}}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Deduce that if \({\text{P}}(X = x) > {\text{P}}(X = x - 1)\) then \(x < (n + 1)p\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Hence, determine the value of <em>x</em> which maximizes \({\text{P}}(X = x)\) when \((n + 1)p\) is not an integer.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Given that <em>n</em> = 19 , find the set of values of <em>p</em> for which <em>X</em> has a unique mode of 13.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Each week the management of a football club recorded the number of injuries suffered by their playing staff in that week. The results for a 52-week period were as follows:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Calculate the mean and variance of the number of injuries per week.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Explain why these values provide supporting evidence for using a Poisson distribution model.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>If \(X\) is a random variable that follows a Poisson distribution with mean \(\lambda > 0\) then the probability generating function of \(X\) is \(G(t) = {e^{\lambda (t - 1)}}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Prove that \({\text{E}}(X) = \lambda \).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Prove that \({\text{Var}}(X) = \lambda \).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">\(Y\) is a random variable, independent of \(X\), that also follows a Poisson distribution with mean \(\lambda \).</p>
<p class="p1">If \(S = 2X - Y\) find</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>\({\text{E}}(S)\);</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>\({\text{Var}}(S)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Let \(T = \frac{Y}{2} + \frac{Y}{2}\).</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>Show that \(T\) is an unbiased estimator for \(\lambda \).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Show that \(T\) is a more efficient unbiased estimator of \(\lambda \) than \(S\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Could either \(S\) or \(T\) model a Poisson distribution? Justify your answer.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">By consideration of the probability generating function, \({G_{X + Y}}(t)\), of \(X + Y\), prove that \(X + Y\) follows a Poisson distribution with mean \(2\lambda \).</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>\({G_{X + Y}}(1)\);</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>\({G_{X + Y}}( - 1)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence find the probability that \(X + Y\) is an even number.</p>
<div class="marks">[3]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Engine oil is sold in cans of two capacities, large and small. The amount, in millilitres, in each can, is normally distributed according to Large \( \sim {\text{N}}(5000,{\text{ }}40)\) and Small \( \sim {\text{N}}(1000,{\text{ }}25)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A large can is selected at random. Find the probability that the can contains at least \(4995\) millilitres of oil.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A large can and a small can are selected at random. Find the probability that the large can contains at least \(30\) milliliters more than five times the amount contained in the small can.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A large can and five small cans are selected at random. Find the probability that the large can contains at least \(30\) milliliters less than the total amount contained in the small cans.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">When Andrew throws a dart at a target, the probability that he hits it is \(\frac{1}{3}\) ; when Bill throws a dart at the target, the probability that he hits the it is \(\frac{1}{4}\) . Successive throws are independent. One evening, they throw darts at the target alternately, starting with Andrew, and stopping as soon as one of their darts hits the target. Let <em>X</em> denote the total number of darts thrown.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the value of \({\text{P}}(X = 1)\) and show that \({\text{P}}(X = 2) = \frac{1}{6}\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that the probability generating function for <em>X</em> is given by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[G(t) = \frac{{2t + {t^2}}}{{6 - 3{t^2}}}.\]</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence determine \({\text{E}}(X)\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The weights of the oranges produced by a farm may be assumed to be normally distributed with mean 205 grams and standard deviation 10 grams.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the probability that a randomly chosen orange weighs more than 200 grams.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Five of these oranges are selected at random to be put into a bag. Find the probability that the combined weight of the five oranges is less than 1 kilogram.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The farm also produces lemons whose weights may be assumed to be normally distributed with mean 75 grams and standard deviation 3 grams. Find the probability that the weight of a randomly chosen orange is more than three times the weight of a randomly chosen lemon.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Determine the probability generating function for \(X \sim {\text{B}}(1,{\text{ }}p)\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Explain why the probability generating function for \({\text{B}}(n,{\text{ }}p)\) is a polynomial of degree \(n\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Two independent random variables \({X_1}\) and \({X_2}\) are such that \({X_1} \sim {\text{B}}(1,{\text{ }}{p_1})\) <span class="s1">and \({X_2} \sim {\text{B}}(1,{\text{ }}{p_2})\)</span>. Prove that if \({X_1} + {X_2}\) has a binomial distribution then \({p_1} = {p_2}\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>X</em> is assumed to have probability density function <em>f</em>, where</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[f(x) = \left\{ {\begin{array}{*{20}{c}}<br> {\frac{x}{{18,}}}&{0 \leqslant x \leqslant 6} \\ <br> {0,}&{{\text{otherwise}}{\text{.}}} <br>\end{array}} \right.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that if the assumption is correct, then</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{\text{P}}(a \leqslant X \leqslant b) = \frac{{{b^2} - {a^2}}}{{36}},{\text{ for }}0 \leqslant a \leqslant b \leqslant 6.\]</span></p>
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<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The discrete random variable <em>X</em> has the following probability distribution, where \(0 < \theta < \frac{1}{3}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 23px/normal Helvetica; text-align: center; margin: 0px;"><img src="data:image/png;base64,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" alt></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine \({\text{E}}(X)\) and show that \({\text{Var}}(X) = 6\theta - 16{\theta ^2}\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In order to estimate \(\theta \), a random sample of <em>n</em> observations is obtained from the distribution of <em>X</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Given that \({\bar X}\) denotes the mean of this sample, show that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{{\hat \theta }_1} = \frac{{3 - \bar X}}{4}\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">is an unbiased estimator for \(\theta \) and write down an expression for the variance of \({{\hat \theta }_1}\) in terms of <em>n</em> and \(\theta \).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Let <em>Y</em> denote the number of observations that are equal to 1 in the sample. Show that <em>Y</em> has the binomial distribution \({\text{B}}(n,{\text{ }}\theta )\) and deduce that \({{\hat \theta }_2} = \frac{Y}{n}\) is another unbiased estimator for \(\theta \). Obtain an expression for the variance of \({{\hat \theta }_2}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Show that \({\text{Var}}({{\hat \theta }_1}) < {\text{Var}}({{\hat \theta }_2})\) and state, with a reason, which is the more </span><span style="font-family: 'times new roman', times; font-size: medium;">efficient estimator, \({{\hat \theta }_1}\) or \({{\hat \theta }_2}\).</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">b.</div>
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<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Bill also has a box with 10 biscuits in it. 4 biscuits are chocolate and 6 are plain. Bill takes a biscuit from his box at random, looks at it and replaces it in the box. He repeats this process until he has looked at 5 biscuits in total. Let <em>B </em>be the number of chocolate biscuits that Bill takes and looks at.</span></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">State the distribution of <em>B </em>.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find P(<em>B </em>= 3) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 36.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find P(<em>B </em>= 5) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
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<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The random variable <em>X </em>has probability distribution Po(8).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find \({\text{P}}(X = 6)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find \({\text{P}}(X = 6|5 \leqslant X \leqslant 8)\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\(\bar X\) denotes the sample mean of \(n > 1\) independent observations from \(X\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Write down \({\text{E}}(\bar X)\) and \({\text{Var}}(\bar X)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence, give a reason why \(\bar X\) is not a Poisson distribution.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">A random sample of \(40\) observations is taken from the distribution for \(X\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find \({\text{P}}(7.1 < \bar X < 8.5)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Given that \({\text{P}}\left( {\left| {\bar X - 8} \right| \leqslant k} \right) = 0.95\), find the value of \(k\).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
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<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The continuous random variable <em>X </em>has probability density function <em>f </em>given by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">\[f(x) = \left\{ {\begin{array}{*{20}{c}}<br> {2x,}&{0 \leqslant x \leqslant 0.5,} \\ <br> {\frac{4}{3} - \frac{2}{3}x,}&{0.5 \leqslant x \leqslant 2} \\ <br> {0,}&{{\text{otherwise}}{\text{.}}} <br>\end{array}} \right.\]</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Sketch the function <em>f </em>and show that the lower quartile is 0.5.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Determine E(<em>X </em>).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Determine \({\text{E}}({X^2})\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Two independent observations are made from <em>X </em>and the values are added.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The resulting random variable is denoted <em>Y </em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Determine \({\text{E}}(Y - 2X)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Determine \({\text{Var}}\,(Y - 2X)\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find the cumulative distribution function for <em>X </em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence, or otherwise, find the median of the distribution.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
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<br><hr><br><div class="specification">
<p class="p1">A random variable \(X\) has probability density function</p>
<p class="p1">\(f(x) = \left\{ {\begin{array}{*{20}{c}} 0&{x < 0} \\ {\frac{1}{2}}&{0 \le x < 1} \\ {\frac{1}{4}}&{1 \le x < 3} \\ 0&{x \ge 3} \end{array}} \right.\)</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of \(y = f(x)\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the cumulative distribution function for \(X\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the interquartile range for \(X\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
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