document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1034101>Question 1034101</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose that x is a random variable with the possible values 1,2,and 3. Given that E(X^2)<=6, find the maxium possible value of E(X).  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #648734 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=648734\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let p(1) = p, p(2) = q, and p(3) = 1 - p - q.\r<BR>\n" );
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document.write( "Now <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E%28X%5E2%29+=+1%5E2%2Ap%2B2%5E2%2Aq%2B3%5E2%2A%281-p-q%29+=+p%2B4q%2B9%281-p-q%29+=+9-8p-5q ALIGN=MIDDLE  ALT=\"E%28X%5E2%29+=+1%5E2%2Ap%2B2%5E2%2Aq%2B3%5E2%2A%281-p-q%29+=+p%2B4q%2B9%281-p-q%29+=+9-8p-5q\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E%28X%5E2%29+%3C=+6 ALIGN=MIDDLE  ALT=\"E%28X%5E2%29+%3C=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, ==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9-8p-5q+%3C=6 ALIGN=MIDDLE  ALT=\"9-8p-5q+%3C=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > ==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8p%2B5q+%3E=+3 ALIGN=MIDDLE  ALT=\"8p%2B5q+%3E=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Also, there are other restrictions, like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C=p%3C=1 ALIGN=MIDDLE  ALT=\"0+%3C=p%3C=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C=q%3C=1 ALIGN=MIDDLE  ALT=\"0+%3C=q%3C=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%2Bq+%3C=1 ALIGN=MIDDLE  ALT=\"p%2Bq+%3C=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E%28X%29+=+1%2Ap%2B2%2Aq%2B3%2A%281-p-q%29+=+p%2B2q%2B3%281-p-q%29+=+3-2p-q ALIGN=MIDDLE  ALT=\"E%28X%29+=+1%2Ap%2B2%2Aq%2B3%2A%281-p-q%29+=+p%2B2q%2B3%281-p-q%29+=+3-2p-q\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "This now becomes a linear programming problem, namely,\r<BR>\n" );
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document.write( "Maximize <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E+=+3-2p-q ALIGN=MIDDLE  ALT=\"E+=+3-2p-q\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <BR>\n" );
document.write( "subject to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8p%2B5q+%3E=+3 ALIGN=MIDDLE  ALT=\"8p%2B5q+%3E=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%2Bq+%3C=1 ALIGN=MIDDLE  ALT=\"p%2Bq+%3C=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C=p%3C=1 ALIGN=MIDDLE  ALT=\"0+%3C=p%3C=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C=q%3C=1 ALIGN=MIDDLE  ALT=\"0+%3C=q%3C=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Letting p be the horizontal axis, and q the vertical axis, using the graphical method, the region of feasibility is found to be a quadrilateral with the corner points (0,3/5), (0,1), (1,0), and (3/8,0).\r<BR>\n" );
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document.write( "Now... <BR>\n" );
document.write( "E(0,3/5) = 2.4,<BR>\n" );
document.write( "E(0,1) = 2,<BR>\n" );
document.write( "E(1,0) = 1, and <BR>\n" );
document.write( "E(3/8,0) = 2.25.\r<BR>\n" );
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document.write( "Therefore the maximum possible value of E(X) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%282.4%29 ALIGN=MIDDLE  ALT=\"highlight%282.4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and happens when p(1) = 0, p(2) = 3/5, and p(3) = 2/5.\r<BR>\n" );
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document.write( "(This answer is also consistent with the upper and lower bounds for E(X).  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Var%28X%29%3E=0 ALIGN=MIDDLE  ALT=\"Var%28X%29%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E%28X%5E2%29+%3E=+%28E%28X%29%29%5E2 ALIGN=MIDDLE  ALT=\"E%28X%5E2%29+%3E=+%28E%28X%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%3E=+%28E%28X%29%29%5E2 ALIGN=MIDDLE  ALT=\"6%3E=+%28E%28X%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%286%29+%3E=+E%28X%29+%3E=+-sqrt%286%29 ALIGN=MIDDLE  ALT=\"sqrt%286%29+%3E=+E%28X%29+%3E=+-sqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%286%29 ALIGN=MIDDLE  ALT=\"sqrt%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is around 2.4495 to 5 significant figures.)\r<BR>\n" );
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