document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1172355>Question 1172355</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A random variable X has pdf = 1 - x/2 in interval (0, 2). What is the expected value of random variable X² </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850823 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850823\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's find the expected value of X².\r<BR>\n" );
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document.write( "**Understanding Expected Value**\r<BR>\n" );
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document.write( "The expected value of a function g(X) of a continuous random variable X with probability density function (PDF) f(x) is given by:\r<BR>\n" );
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document.write( "E[g(X)] = ∫ g(x) * f(x) dx\r<BR>\n" );
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document.write( "**Applying the Formula**\r<BR>\n" );
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document.write( "In this case:\r<BR>\n" );
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document.write( "* g(X) = X²<BR>\n" );
document.write( "* f(x) = 1 - x/2 for 0 ≤ x ≤ 2<BR>\n" );
document.write( "* f(x) = 0 elsewhere\r<BR>\n" );
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document.write( "So, we need to calculate:\r<BR>\n" );
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document.write( "E[X²] = ∫(from 0 to 2) x² * (1 - x/2) dx\r<BR>\n" );
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document.write( "**Calculating the Integral**\r<BR>\n" );
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document.write( "1.  **Expand the expression:**<BR>\n" );
document.write( "    * E[X²] = ∫(from 0 to 2) (x² - x³/2) dx\r<BR>\n" );
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document.write( "2.  **Integrate:**<BR>\n" );
document.write( "    * E[X²] = [x³/3 - x⁴/8](from 0 to 2)\r<BR>\n" );
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document.write( "3.  **Evaluate the integral at the limits:**<BR>\n" );
document.write( "    * E[X²] = [(2)³/3 - (2)⁴/8] - [(0)³/3 - (0)⁴/8]<BR>\n" );
document.write( "    * E[X²] = [8/3 - 16/8] - [0]<BR>\n" );
document.write( "    * E[X²] = 8/3 - 2<BR>\n" );
document.write( "    * E[X²] = 8/3 - 6/3<BR>\n" );
document.write( "    * E[X²] = 2/3\r<BR>\n" );
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document.write( "**Result**\r<BR>\n" );
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document.write( "Therefore, the expected value of X² is 2/3, which is approximately 0.6667.<BR>\n" );
document.write( " </SPAN>\n" );
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