document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177615>Question 1177615</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find E(X) where X is the outcome when one rolls a six-sided balanced die. Find the mgf of X. Also, using the mgf of X, compute the variance of X.\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 #850433 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</B></A><img src=\"/images/stars/stars-13.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=850433\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Absolutely! Let's break down the calculations step by step.\r<BR>\n" );
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document.write( "1. **Expected Value (E(X))**\r<BR>\n" );
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document.write( "The expected value of a discrete random variable is the sum of the products of each possible value and its probability. For a fair six-sided die, each outcome (1 to 6) has a probability of 1/6. Therefore:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "E(X) = (1)(1/6) + (2)(1/6) + (3)(1/6) + (4)(1/6) + (5)(1/6) + (6)(1/6)<BR>\n" );
document.write( "     = 21/6 = 3.5<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "So, the expected value of a single roll is 3.5.\r<BR>\n" );
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document.write( "2. **Moment Generating Function (MGF)**\r<BR>\n" );
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document.write( "The moment generating function of a random variable X is defined as:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "M_X(t) = E(e^(tX))<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "For our die roll:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "M_X(t) = E(e^(tX)) = (1/6)(e^t + e^(2t) + e^(3t) + e^(4t) + e^(5t) + e^(6t))<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "3. **Variance using the MGF**\r<BR>\n" );
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document.write( "The variance can be computed using the MGF as follows:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "Var(X) = E(X^2) - (E(X))^2<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "We can find E(X^2) using the second derivative of the MGF:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "E(X^2) = M_X''(0)<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "Taking the second derivative of M_X(t) and evaluating it at t=0 is a bit tedious, but doing the calculation gives us:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "E(X^2) = 91/6<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "Now we can compute the variance:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "Var(X) = E(X^2) - (E(X))^2 = (91/6) - (3.5)^2 = 35/12<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "Feel free to ask if you have any further questions or would like to explore other properties of the die roll distribution! </SPAN>\n" );
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