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<h2 class="hd hd-2 unit-title">7.1 Generating samples for random variables</h2>
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<h3 class="hd hd-2">Video</h3>
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<h2 class="hd hd-2 unit-title">7.2 Generating samples for random vectors</h2>
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<h3 class="hd hd-2">Video</h3>
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<h2 class="hd hd-2 unit-title">7.3 Integration with Monte Carlo simulation</h2>
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<h2 class="hd hd-2 unit-title">7.4 Failure probability estimation</h2>
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<h2 class="hd hd-2 unit-title">7.5 Importance sampling</h2>
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<h2 class="hd hd-2 unit-title">Chapter 7 Homework</h2>
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<p class="MsoNormal"><b style="mso-bidi-font-weight: normal;"><span style="font-size: 12.0pt; mso-bidi-font-family: 'Times New Roman';">Problem:</span></b><span style="font-size: 12.0pt; mso-bidi-font-family: 'Times New Roman';"> Suppose <i style="mso-bidi-font-style: normal;">g</i>(<i style="mso-bidi-font-style: normal;">x</i><sub>1</sub><i style="mso-bidi-font-style: normal;">, </i></span><i style="mso-bidi-font-style: normal;"><span style="font-size: 12.0pt; mso-fareast-font-family: DengXian; mso-fareast-theme-font: minor-fareast; mso-bidi-font-family: 'Times New Roman';">x</span></i><sub><span style="font-size: 12.0pt; mso-bidi-font-family: 'Times New Roman';">2</span></sub><span style="font-size: 12.0pt; mso-bidi-font-family: 'Times New Roman';">) is a function of two normal variables <i style="mso-bidi-font-style: normal;">x</i><sub>1</sub> and </span><i style="mso-bidi-font-style: normal;"><span style="font-size: 12.0pt; mso-fareast-font-family: DengXian; mso-fareast-theme-font: minor-fareast; mso-bidi-font-family: 'Times New Roman';">x</span></i><sub><span style="font-size: 12.0pt; mso-bidi-font-family: 'Times New Roman';">2</span></sub><span style="font-size: 12.0pt; mso-bidi-font-family: 'Times New Roman';"> as follows:<o:p></o:p></span></p>
<p class="MsoNormal" align="center" style="text-align: center; text-indent: 24.0pt;"><i style="mso-bidi-font-style: normal;"><span style="font-size: 12.0pt; mso-bidi-font-family: 'Times New Roman';"><span style="position: relative; top: 6.0pt; mso-text-raise: -6.0pt;"><v:shapetype id="_x0000_t75" coordsize="21600,21600" o:spt="75" o:preferrelative="t" path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f"> <v:stroke joinstyle="miter"> <v:formulas> <v:f eqn="if lineDrawn pixelLineWidth 0"> <v:f eqn="sum @0 1 0"> <v:f eqn="sum 0 0 @1"> <v:f eqn="prod @2 1 2"> <v:f eqn="prod @3 21600 pixelWidth"> <v:f eqn="prod @3 21600 pixelHeight"> <v:f eqn="sum @0 0 1"> <v:f eqn="prod @6 1 2"> <v:f eqn="prod @7 21600 pixelWidth"> <v:f eqn="sum @8 21600 0"> <v:f eqn="prod @7 21600 pixelHeight"> <v:f eqn="sum @10 21600 0"> </v:f></v:f></v:f></v:f></v:f></v:f></v:f></v:f></v:f></v:f></v:f></v:f></v:formulas> <v:path o:extrusionok="f" gradientshapeok="t" o:connecttype="rect"> <o:lock v:ext="edit" aspectratio="t"> </o:lock></v:path></v:stroke></v:shapetype><v:shape id="_x0000_i1025" type="#_x0000_t75" style="width: 117pt; height: 19.5pt;" o:ole=""> <v:imagedata src="file:///C:/Users/Kostis/AppData/Local/Temp/msohtmlclip1/01/clip_image001.wmz" o:title=""> </v:imagedata></v:shape></span><!-- [if gte mso 9]><xml>
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<p class="MsoNormal" align="center" style="text-align: center;"><span style="font-size: 12.0pt; mso-bidi-font-family: 'Times New Roman';"><o:p> </o:p></span></p>
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<td width="76" style="width: 56.7pt; border: solid windowtext 1.0pt; border-left: none; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 17.0pt;">
<p class="MsoNormal" align="center" style="text-align: center;"><i><span style="font-size: 12.0pt; mso-bidi-font-family: 'Times New Roman';">x<sub>1</sub><o:p></o:p></span></i></p>
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<td width="76" style="width: 56.7pt; border: solid windowtext 1.0pt; border-left: none; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 17.0pt;">
<p class="MsoNormal" align="center" style="text-align: center;"><i><span style="font-size: 12.0pt; mso-fareast-font-family: DengXian; mso-fareast-theme-font: minor-fareast; mso-bidi-font-family: 'Times New Roman';">x<sub>2</sub><o:p></o:p></span></i></p>
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<tr style="mso-yfti-irow: 1; height: 17.0pt;">
<td width="76" style="width: 56.7pt; border: solid windowtext 1.0pt; border-top: none; mso-border-top-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 17.0pt;">
<p class="MsoNormal" align="center" style="text-align: center;"><i><span style="font-size: 12.0pt; font-family: Symbol; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'; mso-bidi-font-family: 'Times New Roman'; mso-char-type: symbol; mso-symbol-font-family: Symbol;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">m</span></span></i><i><span style="font-size: 12.0pt; mso-bidi-font-family: 'Times New Roman';"><o:p></o:p></span></i></p>
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<td width="76" style="width: 56.7pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 17.0pt;">
<p class="MsoNormal" align="center" style="text-align: center;"><span style="font-size: 12.0pt; mso-fareast-font-family: DengXian; mso-fareast-theme-font: minor-fareast; mso-bidi-font-family: 'Times New Roman';">2<o:p></o:p></span></p>
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<td width="76" style="width: 56.7pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 17.0pt;">
<p class="MsoNormal" align="center" style="text-align: center;"><span style="font-size: 12.0pt; mso-fareast-font-family: DengXian; mso-fareast-theme-font: minor-fareast; mso-bidi-font-family: 'Times New Roman';">3<o:p></o:p></span></p>
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<td width="76" style="width: 56.7pt; border: solid windowtext 1.0pt; border-top: none; mso-border-top-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 17.0pt;">
<p class="MsoNormal" align="center" style="text-align: center;"><i><span style="font-size: 12.0pt; font-family: Symbol; mso-ascii-font-family: 'Times New Roman'; mso-hansi-font-family: 'Times New Roman'; mso-bidi-font-family: 'Times New Roman'; mso-char-type: symbol; mso-symbol-font-family: Symbol;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">s</span></span></i><i><span style="font-size: 12.0pt; mso-bidi-font-family: 'Times New Roman';"><o:p></o:p></span></i></p>
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<td width="76" style="width: 56.7pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 17.0pt;">
<p class="MsoNormal" align="center" style="text-align: center;"><span style="font-size: 12.0pt; mso-fareast-font-family: DengXian; mso-fareast-theme-font: minor-fareast; mso-bidi-font-family: 'Times New Roman';">1<o:p></o:p></span></p>
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<td width="76" style="width: 56.7pt; border-top: none; border-left: none; border-bottom: solid windowtext 1.0pt; border-right: solid windowtext 1.0pt; mso-border-top-alt: solid windowtext .5pt; mso-border-left-alt: solid windowtext .5pt; mso-border-alt: solid windowtext .5pt; padding: 0in 5.4pt 0in 5.4pt; height: 17.0pt;">
<p class="MsoNormal" align="center" style="text-align: center;"><span style="font-size: 12.0pt; mso-fareast-font-family: DengXian; mso-fareast-theme-font: minor-fareast; mso-bidi-font-family: 'Times New Roman';">1<o:p></o:p></span></p>
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<p class="MsoNormal" style="text-indent: 24.0pt;"><span style="font-size: 12.0pt; mso-bidi-font-family: 'Times New Roman';">Assume the correlation coefficient between <i style="mso-bidi-font-style: normal;">x</i><sub>1</sub> and </span><i style="mso-bidi-font-style: normal;"><span style="font-size: 12.0pt; mso-fareast-font-family: DengXian; mso-fareast-theme-font: minor-fareast; mso-bidi-font-family: 'Times New Roman';">x</span></i><sub><span style="font-size: 12.0pt; mso-bidi-font-family: 'Times New Roman';">2</span></sub><span style="font-size: 12.0pt; mso-bidi-font-family: 'Times New Roman';"> is zero. Suppose <i>g</i>(<i style="mso-bidi-font-style: normal;">x</i><sub>1</sub><i style="mso-bidi-font-style: normal;">, </i></span><i style="mso-bidi-font-style: normal;"><span style="font-size: 12.0pt; mso-fareast-font-family: DengXian; mso-fareast-theme-font: minor-fareast; mso-bidi-font-family: 'Times New Roman';">x</span></i><sub><span style="font-size: 12.0pt; mso-bidi-font-family: 'Times New Roman';">2</span></sub><span style="font-size: 12.0pt; mso-bidi-font-family: 'Times New Roman';">) < 0 denotes failure. Calculate the failure probability of this problem based on MCS with Excel using 10 samples, 100 samples, 1000 samples, and 10000 samples. <o:p></o:p></span></p>
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