document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197966>Question 1197966</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The first three terms of the power series expansion of the moment generating<BR>\n" );
document.write( "function of the random variable X are 1−t+<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+t%5E2+ ALIGN=MIDDLE  ALT=\"+t%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >. What are the first three terms in the power<BR>\n" );
document.write( "series expansion of the moment generating function of the random variable Y = 1 − X?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848352 by </B><A HREF=/tutors/aboutme.mpl?userid=onyulee><B>onyulee(41)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=onyulee><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=848352\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Find the MGF of Y**\r<BR>\n" );
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document.write( "* **Definition of MGF for Y:**<BR>\n" );
document.write( "   * M_Y(t) = E[e^(tY)] <BR>\n" );
document.write( "   * Since Y = 1 - X, we have:<BR>\n" );
document.write( "      * M_Y(t) = E[e^(t(1-X))] <BR>\n" );
document.write( "      * M_Y(t) = E[e^t * e^(-tX)] <BR>\n" );
document.write( "      * M_Y(t) = e^t * E[e^(-tX)] \r<BR>\n" );
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document.write( "* **Recognize E[e^(-tX)]** <BR>\n" );
document.write( "   * E[e^(-tX)] is the MGF of -X, which is M_X(-t)\r<BR>\n" );
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document.write( "* **Therefore:**<BR>\n" );
document.write( "   * M_Y(t) = e^t * M_X(-t)\r<BR>\n" );
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document.write( "**2. Find the first three terms of M_X(-t)**\r<BR>\n" );
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document.write( "* Given M_X(t) = 1 - t + t^2, we can find M_X(-t) by substituting -t for t:<BR>\n" );
document.write( "   * M_X(-t) = 1 - (-t) + (-t)^2 <BR>\n" );
document.write( "   * M_X(-t) = 1 + t + t^2\r<BR>\n" );
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document.write( "**3. Find the first three terms of M_Y(t)**\r<BR>\n" );
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document.write( "* M_Y(t) = e^t * M_X(-t)<BR>\n" );
document.write( "* M_Y(t) = e^t * (1 + t + t^2)\r<BR>\n" );
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document.write( "* **Recall the Taylor series expansion of e^t:**<BR>\n" );
document.write( "   * e^t = 1 + t + (t^2)/2! + (t^3)/3! + ... \r<BR>\n" );
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document.write( "* **Multiply e^t with (1 + t + t^2) and consider only the first three terms:**<BR>\n" );
document.write( "   * M_Y(t) ≈ (1 + t + (t^2)/2) * (1 + t + t^2) <BR>\n" );
document.write( "   * M_Y(t) ≈ 1 + t + t^2 + t + t^2 + (t^2)/2 <BR>\n" );
document.write( "   * M_Y(t) ≈ 1 + 2t + (5/2)t^2 \r<BR>\n" );
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document.write( "**Therefore, the first three terms in the power series expansion of the moment generating function of the random variable Y = 1 - X are 1 + 2t + (5/2)t^2.**<BR>\n" );
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