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<h2 class="hd hd-2 unit-title">4.1 Linear function of uncertain variables</h2>
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<h3 class="hd hd-2">Video</h3>
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<h2 class="hd hd-2 unit-title">4.2 Taylor series expansion</h2>
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<h3 class="hd hd-2">Video</h3>
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<h2 class="hd hd-2 unit-title">4.3 Point estimate method</h2>
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<h3 class="hd hd-2">Video</h3>
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<h2 class="hd hd-2 unit-title">Chapter 4 Homework</h2>
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<p style="margin: 0in; text-align: justify; text-justify: inter-ideograph;"><b><span style="font-family: 'Times New Roman',serif;">Problem: </span></b><span style="font-family: 'Times New Roman',serif; mso-bidi-font-weight: bold;">Suppose </span><i style="mso-bidi-font-style: normal;"><span style="font-family: 'Times New Roman',serif; mso-fareast-font-family: DengXian; mso-bidi-font-weight: bold;">g</span></i><span style="font-family: 'Times New Roman',serif; mso-fareast-font-family: DengXian; mso-bidi-font-weight: bold;">(<i style="mso-bidi-font-style: normal;">x</i>, <i style="mso-bidi-font-style: normal;">y</i>)</span><span style="font-family: 'Times New Roman',serif; mso-bidi-font-weight: bold;"> is a function of <i style="mso-bidi-font-style: normal;">x</i> and <i style="mso-bidi-font-style: normal;">y</i> as follows:<o:p></o:p></span></p>
<p align="center" style="margin: 0in; text-align: center;"><i style="mso-bidi-font-style: normal;"><span style="font-family: 'Times New Roman',serif; mso-fareast-font-family: DengXian; mso-bidi-font-weight: bold;">g</span></i><span style="font-family: 'Times New Roman',serif; mso-fareast-font-family: DengXian; mso-bidi-font-weight: bold;">(<i style="mso-bidi-font-style: normal;">x</i>, <i style="mso-bidi-font-style: normal;">y</i>) = <i style="mso-bidi-font-style: normal;">x</i><sup>2</sup>+<i style="mso-bidi-font-style: normal;">xy</i>-<i style="mso-bidi-font-style: normal;">y</i><o:p></o:p></span></p>
<p style="margin: 0in; text-align: justify; text-justify: inter-ideograph;"><span style="font-family: 'Times New Roman',serif; mso-fareast-font-family: DengXian; mso-bidi-font-weight: bold;">where the mean and standard deviation values of <i>x</i> and <i>y</i> are:<o:p></o:p></span></p>
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<p align="center" style="margin: 0in; text-align: center;"><span style="font-family: 'Times New Roman',serif; mso-fareast-font-family: DengXian;"><o:p> </o:p></span></p>
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<p align="center" style="margin: 0in; text-align: center;"><i style="mso-bidi-font-style: normal;"><span style="font-family: 'Times New Roman',serif; mso-fareast-font-family: DengXian;">x<o:p></o:p></span></i></p>
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<p align="center" style="margin: 0in; text-align: center;"><i style="mso-bidi-font-style: normal;"><span style="font-family: 'Times New Roman',serif; mso-fareast-font-family: DengXian;">y<o:p></o:p></span></i></p>
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<p align="center" style="margin: 0in; text-align: center;"><span style="font-family: 'Times New Roman',serif; mso-fareast-font-family: DengXian;">Mean<o:p></o:p></span></p>
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<p align="center" style="margin: 0in; text-align: center;"><span style="font-family: 'Times New Roman',serif; mso-fareast-font-family: DengXian;">2<o:p></o:p></span></p>
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<p align="center" style="margin: 0in; text-align: center;"><span style="font-family: 'Times New Roman',serif; mso-fareast-font-family: DengXian;">3<o:p></o:p></span></p>
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<p align="center" style="margin: 0in; text-align: center;"><span style="font-family: 'Times New Roman',serif; mso-fareast-font-family: DengXian;">Std.<o:p></o:p></span></p>
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<p align="center" style="margin: 0in; text-align: center;"><span style="font-family: 'Times New Roman',serif; mso-fareast-font-family: DengXian;">1<o:p></o:p></span></p>
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<p align="center" style="margin: 0in; text-align: center;"><span style="font-family: 'Times New Roman',serif; mso-fareast-font-family: DengXian;">1<o:p></o:p></span></p>
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<p style="margin: 0in; text-align: justify; text-justify: inter-ideograph; text-indent: 24.0pt; mso-char-indent-count: 2.0;"><span style="font-family: 'Times New Roman',serif; mso-fareast-font-family: DengXian; mso-bidi-font-weight: bold;">Assume the correlation coefficient between <i>x</i> and <i>y</i> is -0.5. Answer the following questions.<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left: 0in; text-indent: 24.0pt; mso-char-indent-count: 2.0; mso-list: l0 level1 lfo1;"><!-- [if !supportLists]--><span style="mso-list: Ignore;">(1)<span style="font: 7.0pt 'Times New Roman';"> </span></span><!--[endif]-->Determine the mean and standard deviation of <i style="mso-bidi-font-style: normal;"><span style="mso-fareast-font-family: DengXian; mso-bidi-font-weight: bold;">g</span></i><span style="mso-fareast-font-family: DengXian; mso-bidi-font-weight: bold;">(<i style="mso-bidi-font-style: normal;">x</i>, <i style="mso-bidi-font-style: normal;">y</i>)</span> using the Taylor series expansion method and the Rosenblueth’s method, respectively;<o:p></o:p></p>
<p class="MsoNormal" style="margin-left: 0in; text-indent: 24.0pt; mso-char-indent-count: 2.0; mso-list: l0 level1 lfo1;"><!-- [if !supportLists]--><span style="mso-list: Ignore;">(2)<span style="font: 7.0pt 'Times New Roman';"> </span></span><!--[endif]-->Suppose <i style="mso-bidi-font-style: normal;"><span style="mso-fareast-font-family: DengXian; mso-bidi-font-weight: bold;">g</span></i><span style="mso-fareast-font-family: DengXian; mso-bidi-font-weight: bold;">(<i style="mso-bidi-font-style: normal;">x</i>, <i style="mso-bidi-font-style: normal;">y</i>) </span>< 0 denote failure. What is the failure probability based on the Taylor series expansion method?<o:p></o:p></p>
<p class="MsoNormal" style="margin-left: 0in; text-indent: 24.0pt; mso-char-indent-count: 2.0; mso-list: l0 level1 lfo1;"><!-- [if !supportLists]--><span style="mso-list: Ignore;">(3)<span style="font: 7.0pt 'Times New Roman';"> </span></span><!--[endif]-->Suppose <i style="mso-bidi-font-style: normal;"><span style="mso-fareast-font-family: DengXian; mso-bidi-font-weight: bold;">g</span></i><span style="mso-fareast-font-family: DengXian; mso-bidi-font-weight: bold;">(<i style="mso-bidi-font-style: normal;">x</i>, <i style="mso-bidi-font-style: normal;">y</i>) </span>< 1 denote failure. What is the failure probability based on the Taylor series expansion method?<o:p></o:p></p>
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