document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207202>Question 1207202</A>: <I><SPAN STYLE=\"word-break: break-all;\"> . Independent random variables<BR>\n" );
document.write( "X<BR>\n" );
document.write( "and<BR>\n" );
document.write( "Y<BR>\n" );
document.write( "are such that<BR>\n" );
document.write( "E X( ) 4  , E Y( ) 5  , Var X( ) 1  ,<BR>\n" );
document.write( "Var Y( ) 2 <BR>\n" );
document.write( ". Find:<BR>\n" );
document.write( "A.<BR>\n" );
document.write( "E X Y (4 2 )  .<BR>\n" );
document.write( "B.<BR>\n" );
document.write( "E X Y (5 )  .<BR>\n" );
document.write( "C.<BR>\n" );
document.write( "Var X Y (3 2 )  .<BR>\n" );
document.write( "D.<BR>\n" );
document.write( "Var Y X (5 3 )  . <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847942 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</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=847942\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Understanding the Problem:\r<BR>\n" );
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document.write( "We are given two independent random variables, X and Y, with their respective means and variances:\r<BR>\n" );
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document.write( "E[X] = 4<BR>\n" );
document.write( "E[Y] = 5<BR>\n" );
document.write( "Var[X] = 1<BR>\n" );
document.write( "Var[Y] = 2<BR>\n" );
document.write( "Key Property of Independent Random Variables:\r<BR>\n" );
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document.write( "For independent random variables X and Y, the following properties hold:\r<BR>\n" );
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document.write( "Expected Value of the Sum: E[X + Y] = E[X] + E[Y]<BR>\n" );
document.write( "Variance of the Sum: Var[X + Y] = Var[X] + Var[Y]<BR>\n" );
document.write( "Expected Value of the Product: E[XY] = E[X] * E[Y]<BR>\n" );
document.write( "Variance of the Product: Var(XY) = E[X^2]E[Y^2] - (E[X]E[Y])^2<BR>\n" );
document.write( "Solving the Problem:\r<BR>\n" );
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document.write( "A. E[4X + 2Y] = 4E[X] + 2E[Y] = 44 + 25 = 26\r<BR>\n" );
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document.write( "B. E[5X - Y] = 5E[X] - E[Y] = 5*4 - 5 = 15\r<BR>\n" );
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document.write( "C. Var[3X + 2Y] = 9Var[X] + 4Var[Y] = 91 + 42 = 17\r<BR>\n" );
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document.write( "D. Var[5Y - 3X] = 25Var[Y] + 9Var[X] = 252 + 91 = 61\r<BR>\n" );
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document.write( "Therefore, the answers are:\r<BR>\n" );
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document.write( "A. 26<BR>\n" );
document.write( "B. 15<BR>\n" );
document.write( "C. 17<BR>\n" );
document.write( "D. 61 </SPAN>\n" );
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