document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203892>Question 1203892</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let X be a continuous random variable with density function<BR>\n" );
document.write( "f(x) = { theta*x + 3/2*theta^{3/2}*x^2, for 0 < x < 1/sqrt( theta ); 0, otherwise }<BR>\n" );
document.write( "where theta > 0. What is the expected value of X?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #839870 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=839870\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "To find the expected value, we apply the integral to x*f(x) over the limits from -infinity to infinity.\r<BR>\n" );
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document.write( "E(x) = expected value<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large E(\text{x}) = \int_{-\infty}^{\infty}\text{x}*f(\text{x})d\text{x}\">\r<BR>\n" );
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document.write( "Because much of the real number line makes f(x) = 0, we only need to worry about the interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C+x+%3C+1%2F%28sqrt%28theta%29%29 ALIGN=MIDDLE  ALT=\"0+%3C+x+%3C+1%2F%28sqrt%28theta%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is the same as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C+x+%3C+sqrt%281%2Ftheta%29 ALIGN=MIDDLE  ALT=\"0+%3C+x+%3C+sqrt%281%2Ftheta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > where theta > 0\r<BR>\n" );
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document.write( "The goal has been reduced to computing <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large E(\text{x}) = \int_{0}^{\sqrt{1/\theta}}\text{x}*f(\text{x})d\text{x}\">\r<BR>\n" );
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document.write( "Using fairly elementary calculus integration rules, you should find that:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large g(\text{x}) = \int \text{x}*f(\text{x})d\text{x}\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large g(\text{x}) = \int \text{x}*(\theta*\text{x} + (3/2)\theta^{3/2}\text{x}^2)d\text{x}\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large g(\text{x}) = \int (\theta*\text{x}^2 + (3/2)\theta^{3/2}x^3)d\text{x}\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large g(\text{x}) = (1/3)\theta*\text{x}^3 + (3/8)\theta^{3/2}\text{x}^4+C\"><BR>\n" );
document.write( "For some constant C.\r<BR>\n" );
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document.write( "Then,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large g(\text{x}) = (1/3)\theta*\text{x}^3 + (3/8)\theta^{3/2}\text{x}^4+C\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large g(0) = (1/3)\theta*0^3 + (3/8)\theta^{3/2}*0^4+C\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large g(0) = C\"><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large g(\text{x}) = (1/3)\theta*\text{x}^3 + (3/8)\theta^{3/2}\text{x}^4+C\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large g(1/\sqrt{\theta}) = (1/3)\theta*(1/\sqrt{\theta})^3 + (3/8)\theta^{3/2}*(1/\sqrt{\theta})^4+C\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large g(1/\sqrt{\theta}) = (1/3)\theta*\theta^{-3/2} + (3/8)\theta^{3/2}*\theta^{-2}+C\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large g(1/\sqrt{\theta}) = (1/3)\theta^{-1/2} + (3/8)\theta^{-1/2}+C\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large g(1/\sqrt{\theta}) = (1/3+3/8)\theta^{-1/2}+C\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large g(1/\sqrt{\theta}) = (17/24)\theta^{-1/2}+C\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large g(1/\sqrt{\theta}) = C + \frac{17}{24\sqrt{\theta}}\">\r<BR>\n" );
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document.write( "Subtract the two results to find <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large E(\text{x}) = \int_{0}^{\sqrt{1/\theta}}\text{x}*f(\text{x})d\text{x}\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large E(\text{x}) = g(1/\sqrt{\theta}) - g(0)\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large E(\text{x}) = C + \frac{17}{24\sqrt{\theta}} - C\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large E(\text{x}) = \frac{17}{24\sqrt{\theta}}\">\r<BR>\n" );
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document.write( "Therefore, the expected value is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{17}{24\sqrt{\theta}}\"> where theta > 0<BR>\n" );
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