document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177292>Question 1177292</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" );
document.write( "\n" );
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 #850481 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=850481\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step-by-step.\r<BR>\n" );
document.write( "\n" );
document.write( "**1. Find E(X)**\r<BR>\n" );
document.write( "\n" );
document.write( "* X represents the outcome of rolling a six-sided balanced die.<BR>\n" );
document.write( "* Possible values of X: 1, 2, 3, 4, 5, 6<BR>\n" );
document.write( "* Probability of each outcome: 1/6\r<BR>\n" );
document.write( "\n" );
document.write( "The expected value of X is:\r<BR>\n" );
document.write( "\n" );
document.write( "* E(X) = Σ [x * P(X = x)]<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( "* E(X) = (1/6) * (1 + 2 + 3 + 4 + 5 + 6)<BR>\n" );
document.write( "* E(X) = (1/6) * 21<BR>\n" );
document.write( "* E(X) = 21/6 = 7/2 = 3.5\r<BR>\n" );
document.write( "\n" );
document.write( "**2. Find the Moment Generating Function (MGF) of X**\r<BR>\n" );
document.write( "\n" );
document.write( "The MGF of a discrete random variable X is:\r<BR>\n" );
document.write( "\n" );
document.write( "* MX(t) = E(e^(tX)) = Σ [e^(tx) * P(X = x)]\r<BR>\n" );
document.write( "\n" );
document.write( "In our case:\r<BR>\n" );
document.write( "\n" );
document.write( "* MX(t) = (1/6) * (e^t + e^(2t) + e^(3t) + e^(4t) + e^(5t) + e^(6t))\r<BR>\n" );
document.write( "\n" );
document.write( "**3. Compute the Variance of X Using the MGF**\r<BR>\n" );
document.write( "\n" );
document.write( "1.  **Find MX'(t):**\r<BR>\n" );
document.write( "\n" );
document.write( "    * MX'(t) = (1/6) * (e^t + 2e^(2t) + 3e^(3t) + 4e^(4t) + 5e^(5t) + 6e^(6t))\r<BR>\n" );
document.write( "\n" );
document.write( "2.  **Find MX''(t):**\r<BR>\n" );
document.write( "\n" );
document.write( "    * MX''(t) = (1/6) * (e^t + 4e^(2t) + 9e^(3t) + 16e^(4t) + 25e^(5t) + 36e^(6t))\r<BR>\n" );
document.write( "\n" );
document.write( "3.  **Find E(X) and E(X²) using the MGF:**\r<BR>\n" );
document.write( "\n" );
document.write( "    * E(X) = MX'(0) = (1/6) * (1 + 2 + 3 + 4 + 5 + 6) = 21/6 = 7/2 = 3.5<BR>\n" );
document.write( "    * E(X²) = MX''(0) = (1/6) * (1 + 4 + 9 + 16 + 25 + 36) = 91/6\r<BR>\n" );
document.write( "\n" );
document.write( "4.  **Calculate Var(X):**\r<BR>\n" );
document.write( "\n" );
document.write( "    * Var(X) = E(X²) - [E(X)]²<BR>\n" );
document.write( "    * Var(X) = 91/6 - (7/2)²<BR>\n" );
document.write( "    * Var(X) = 91/6 - 49/4<BR>\n" );
document.write( "    * Var(X) = (182 - 147) / 12<BR>\n" );
document.write( "    * Var(X) = 35/12\r<BR>\n" );
document.write( "\n" );
document.write( "**Answers**\r<BR>\n" );
document.write( "\n" );
document.write( "* E(X) = 3.5<BR>\n" );
document.write( "* MX(t) = (1/6) * (e^t + e^(2t) + e^(3t) + e^(4t) + e^(5t) + e^(6t))<BR>\n" );
document.write( "* Var(X) = 35/12<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );