document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177402>Question 1177402</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let X be a random variable with pdf f(x) = ax^2 + b, 0 ≤ x ≤ 1. Find a and b such that E(X) = 5/8\r<BR>\n" );
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document.write( "Thank you..  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850467 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=850467\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem step-by-step.\r<BR>\n" );
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document.write( "**1. Determine the Relationship Between a and b**\r<BR>\n" );
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document.write( "Since f(x) is a probability density function (pdf), the integral of f(x) over its domain must equal 1:\r<BR>\n" );
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document.write( "* ∫[0, 1] (ax² + b) dx = 1\r<BR>\n" );
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document.write( "Integrate:\r<BR>\n" );
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document.write( "* [ (ax³/3) + bx ] from 0 to 1 = 1<BR>\n" );
document.write( "* (a/3 + b) - (0) = 1<BR>\n" );
document.write( "* a/3 + b = 1<BR>\n" );
document.write( "* a + 3b = 3<BR>\n" );
document.write( "* a = 3 - 3b\r<BR>\n" );
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document.write( "**2. Calculate the Expected Value E(X)**\r<BR>\n" );
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document.write( "The expected value of X is given by:\r<BR>\n" );
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document.write( "* E(X) = ∫[0, 1] x * f(x) dx<BR>\n" );
document.write( "* E(X) = ∫[0, 1] x * (ax² + b) dx<BR>\n" );
document.write( "* E(X) = ∫[0, 1] (ax³ + bx) dx\r<BR>\n" );
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document.write( "Integrate:\r<BR>\n" );
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document.write( "* E(X) = [ (ax⁴/4) + (bx²/2) ] from 0 to 1<BR>\n" );
document.write( "* E(X) = (a/4 + b/2) - (0)<BR>\n" );
document.write( "* E(X) = a/4 + b/2\r<BR>\n" );
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document.write( "**3. Use the Given E(X) = 5/8**\r<BR>\n" );
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document.write( "We are given that E(X) = 5/8. So:\r<BR>\n" );
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document.write( "* a/4 + b/2 = 5/8\r<BR>\n" );
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document.write( "Multiply both sides by 8 to eliminate fractions:\r<BR>\n" );
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document.write( "* 2a + 4b = 5\r<BR>\n" );
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document.write( "**4. Substitute a = 3 - 3b**\r<BR>\n" );
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document.write( "Substitute a = 3 - 3b into the equation 2a + 4b = 5:\r<BR>\n" );
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document.write( "* 2(3 - 3b) + 4b = 5<BR>\n" );
document.write( "* 6 - 6b + 4b = 5<BR>\n" );
document.write( "* 6 - 2b = 5<BR>\n" );
document.write( "* -2b = -1<BR>\n" );
document.write( "* b = 1/2\r<BR>\n" );
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document.write( "**5. Find a**\r<BR>\n" );
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document.write( "Substitute b = 1/2 back into a = 3 - 3b:\r<BR>\n" );
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document.write( "* a = 3 - 3(1/2)<BR>\n" );
document.write( "* a = 3 - 3/2<BR>\n" );
document.write( "* a = 6/2 - 3/2<BR>\n" );
document.write( "* a = 3/2\r<BR>\n" );
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document.write( "**Therefore, a = 3/2 and b = 1/2.**<BR>\n" );
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