document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191144>Question 1191144</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find some means. Suppose that X is a random variable with mean 20 and standard deviation 2. Also suppose that Y is a random variable with mean 40 and standard deviation 7. Assume that the correlation between X and Y is zero. Find the mean of the random variable Z for each of the following cases. Be sure to show your work.\r<BR>\n" );
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document.write( "A)Z=25−12X.<BR>\n" );
document.write( "B)Z=13X−8.<BR>\n" );
document.write( "C)Z=X+Y.<BR>\n" );
document.write( "D)Z=X−Y.<BR>\n" );
document.write( "E)Z=−3X+3Y<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 #849170 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=849170\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to calculate the mean (expected value) of Z for each case, given E(X) = 20 and E(Y) = 40:\r<BR>\n" );
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document.write( "**Key Principle:**\r<BR>\n" );
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document.write( "E(aX + bY + c) = aE(X) + bE(Y) + c, where a, b, and c are constants.\r<BR>\n" );
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document.write( "**A) Z = 25 - (1/2)X**\r<BR>\n" );
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document.write( "E(Z) = E(25 - (1/2)X)<BR>\n" );
document.write( "E(Z) = 25 - (1/2)E(X)<BR>\n" );
document.write( "E(Z) = 25 - (1/2)(20)<BR>\n" );
document.write( "E(Z) = 25 - 10<BR>\n" );
document.write( "E(Z) = 15\r<BR>\n" );
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document.write( "**B) Z = (1/3)X - 8**\r<BR>\n" );
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document.write( "E(Z) = E((1/3)X - 8)<BR>\n" );
document.write( "E(Z) = (1/3)E(X) - 8<BR>\n" );
document.write( "E(Z) = (1/3)(20) - 8<BR>\n" );
document.write( "E(Z) = 20/3 - 24/3<BR>\n" );
document.write( "E(Z) = -4/3  (approximately -1.33)\r<BR>\n" );
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document.write( "**C) Z = X + Y**\r<BR>\n" );
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document.write( "E(Z) = E(X + Y)<BR>\n" );
document.write( "E(Z) = E(X) + E(Y)<BR>\n" );
document.write( "E(Z) = 20 + 40<BR>\n" );
document.write( "E(Z) = 60\r<BR>\n" );
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document.write( "**D) Z = X - Y**\r<BR>\n" );
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document.write( "E(Z) = E(X - Y)<BR>\n" );
document.write( "E(Z) = E(X) - E(Y)<BR>\n" );
document.write( "E(Z) = 20 - 40<BR>\n" );
document.write( "E(Z) = -20\r<BR>\n" );
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document.write( "**E) Z = -3X + 3Y**\r<BR>\n" );
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document.write( "E(Z) = E(-3X + 3Y)<BR>\n" );
document.write( "E(Z) = -3E(X) + 3E(Y)<BR>\n" );
document.write( "E(Z) = -3(20) + 3(40)<BR>\n" );
document.write( "E(Z) = -60 + 120<BR>\n" );
document.write( "E(Z) = 60<BR>\n" );
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