document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=349587>Question 349587</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I have a homework question that has me stumped.  Desperatedly need help.<BR>\n" );
document.write( "A random variable X has moment generating function M(t) = (1-t)^-1.  Find the rth moment 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 #249871 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=249871\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M%28t%29=1%2F%281-t%29 ALIGN=MIDDLE  ALT=\"M%28t%29=1%2F%281-t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Look at the derivatives of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M%28t%29 ALIGN=MIDDLE  ALT=\"M%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > evaluated when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0 ALIGN=MIDDLE  ALT=\"t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The rth moment is the rth derivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M%28t%29 ALIGN=MIDDLE  ALT=\"M%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > evaluated at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0 ALIGN=MIDDLE  ALT=\"t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "1st derivative:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M1%28t%29=%281-t%29%5E%28-2%29 ALIGN=MIDDLE  ALT=\"M1%28t%29=%281-t%29%5E%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M1%280%29=1 ALIGN=MIDDLE  ALT=\"M1%280%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "2nd derivative:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M2%28t%29=2%281-t%29%5E%28-3%29 ALIGN=MIDDLE  ALT=\"M2%28t%29=2%281-t%29%5E%28-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M2%280%29=2 ALIGN=MIDDLE  ALT=\"M2%280%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "3rd derivative:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M3%28t%29=6%281-t%29%5E%28-4%29 ALIGN=MIDDLE  ALT=\"M3%28t%29=6%281-t%29%5E%28-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M3%280%29=6 ALIGN=MIDDLE  ALT=\"M3%280%29=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "4th derivative:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M4%28t%29=24%281-t%29%5E%28-5%29 ALIGN=MIDDLE  ALT=\"M4%28t%29=24%281-t%29%5E%28-5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M4%280%29=24 ALIGN=MIDDLE  ALT=\"M4%280%29=24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "5th derivative:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M5%28t%29=120%281-t%29%5E%28-6%29 ALIGN=MIDDLE  ALT=\"M5%28t%29=120%281-t%29%5E%28-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M5%280%29=120 ALIGN=MIDDLE  ALT=\"M5%280%29=120\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Hopefully you see the pattern here,<BR>\n" );
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document.write( "1st derivative: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1=1%21 ALIGN=MIDDLE  ALT=\"1=1%21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "2nd derivative: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2A2=2%21 ALIGN=MIDDLE  ALT=\"1%2A2=2%21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "3rd derivative: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2A2%2A3=3%21 ALIGN=MIDDLE  ALT=\"1%2A2%2A3=3%21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "4th derivative: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2A2%2A3%2A4=4%21 ALIGN=MIDDLE  ALT=\"1%2A2%2A3%2A4=4%21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "5th derivative: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2A2%2A3%2A4%2A5=5%21 ALIGN=MIDDLE  ALT=\"1%2A2%2A3%2A4%2A5=5%21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "rth derivative: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2A2%2A3%2A4%2A5 ALIGN=MIDDLE  ALT=\"1%2A2%2A3%2A4%2A5\"   DISABLED_style= \"max-width:90%; height:auto;\" >*...*<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r-1%29%2Ar ALIGN=MIDDLE  ALT=\"%28r-1%29%2Ar\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%21 ALIGN=MIDDLE  ALT=\"r%21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So the rth moment of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=M%28t%29 ALIGN=MIDDLE  ALT=\"M%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%21 ALIGN=MIDDLE  ALT=\"r%21\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );