document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1176040>Question 1176040</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A car dealership sells 0, 1 or 2 luxury vehicles each day. When selling a car, the dealer also tries to persuade the customer to buy an extended warranty for the car.\r<BR>\n" );
document.write( "\n" );
document.write( "Let X denote the number of luxury cars sold in a given day. Let Y denote the number of extended warranties sold.<BR>\n" );
document.write( "X	Y	Probability<BR>\n" );
document.write( "0	0	0.07<BR>\n" );
document.write( "1	0	0.13<BR>\n" );
document.write( "1	1	0.1<BR>\n" );
document.write( "2	0	0.15<BR>\n" );
document.write( "2	1	0.11<BR>\n" );
document.write( "2	2	0.44<BR>\n" );
document.write( "Use the table to find the variance 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 #850597 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.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=850597\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to calculate the variance of X using the given table:\r<BR>\n" );
document.write( "\n" );
document.write( "**1. Calculate the Probabilities for X**\r<BR>\n" );
document.write( "\n" );
document.write( "* P(X = 0) = 0.07<BR>\n" );
document.write( "* P(X = 1) = 0.13 + 0.10 = 0.23<BR>\n" );
document.write( "* P(X = 2) = 0.15 + 0.44 = 0.59\r<BR>\n" );
document.write( "\n" );
document.write( "**2. Calculate the Expected Value (Mean) of X**\r<BR>\n" );
document.write( "\n" );
document.write( "* E(X) = Σ [x * P(X = x)]<BR>\n" );
document.write( "* E(X) = (0 * 0.07) + (1 * 0.23) + (2 * 0.59)<BR>\n" );
document.write( "* E(X) = 0 + 0.23 + 1.18<BR>\n" );
document.write( "* E(X) = 1.41\r<BR>\n" );
document.write( "\n" );
document.write( "**3. Calculate the Variance of X**\r<BR>\n" );
document.write( "\n" );
document.write( "* Var(X) = E(X²) - [E(X)]²<BR>\n" );
document.write( "* First, calculate E(X²):<BR>\n" );
document.write( "    * E(X²) = Σ [x² * P(X = x)]<BR>\n" );
document.write( "    * E(X²) = (0² * 0.07) + (1² * 0.23) + (2² * 0.59)<BR>\n" );
document.write( "    * E(X²) = 0 + 0.23 + 2.36<BR>\n" );
document.write( "    * E(X²) = 2.59<BR>\n" );
document.write( "* Now, calculate Var(X):<BR>\n" );
document.write( "    * Var(X) = E(X²) - [E(X)]²<BR>\n" );
document.write( "    * Var(X) = 2.59 - (1.41)²<BR>\n" );
document.write( "    * Var(X) = 2.59 - 1.9881<BR>\n" );
document.write( "    * Var(X) = 0.6019\r<BR>\n" );
document.write( "\n" );
document.write( "**Alternative Method**\r<BR>\n" );
document.write( "\n" );
document.write( "* Var(X) = Σ [(x - E(X))² * P(X = x)]<BR>\n" );
document.write( "* Var(X) = [(0 - 1.41)² * 0.07] + [(1 - 1.41)² * 0.23] + [(2 - 1.41)² * 0.59]<BR>\n" );
document.write( "* Var(X) = [(1.9881) * 0.07] + [(0.1681) * 0.23] + [(0.3481) * 0.59]<BR>\n" );
document.write( "* Var(X) = 0.139167 + 0.038663 + 0.205379<BR>\n" );
document.write( "* Var(X) = 0.383209\r<BR>\n" );
document.write( "\n" );
document.write( "**Therefore, the variance of X is 0.383209**<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );