document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191018>Question 1191018</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find some variances and standard deviations. Suppose that X is a random variable with mean 20 and standarddeviation 3. Also suppose that Y is a random variable with mean 60 and standard deviation 2. Assume that the correlation between X and Y is zero. Find the variance and the standard deviation 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=33−8X.<BR>\n" );
document.write( "B)Z=11X−6.<BR>\n" );
document.write( "C)Z=X+Y.<BR>\n" );
document.write( "D)Z=X−Y.<BR>\n" );
document.write( "E)Z=−2X+2Y. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849153 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=849153\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's the breakdown of how to calculate the variance and standard deviation of Z for each case:\r<BR>\n" );
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document.write( "**Key Principles:**\r<BR>\n" );
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document.write( "* **Variance of a constant:** Var(c) = 0<BR>\n" );
document.write( "* **Variance of a scalar multiple:** Var(aX) = a²Var(X)<BR>\n" );
document.write( "* **Variance of a sum/difference (independent variables):** If X and Y are independent (correlation is 0), Var(X + Y) = Var(X) + Var(Y) and Var(X - Y) = Var(X) + Var(Y)<BR>\n" );
document.write( "* **Standard Deviation:** SD(Z) = sqrt(Var(Z))\r<BR>\n" );
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document.write( "We're given Var(X) = 3² = 9 and Var(Y) = 2² = 4.\r<BR>\n" );
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document.write( "**A) Z = 33 - 8X**\r<BR>\n" );
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document.write( "* Var(Z) = Var(33 - 8X) = (-8)²Var(X) = 64 * 9 = 576<BR>\n" );
document.write( "* SD(Z) = sqrt(576) = 24\r<BR>\n" );
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document.write( "**B) Z = 11X - 6**\r<BR>\n" );
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document.write( "* Var(Z) = Var(11X - 6) = 11²Var(X) = 121 * 9 = 1089<BR>\n" );
document.write( "* SD(Z) = sqrt(1089) = 33\r<BR>\n" );
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document.write( "**C) Z = X + Y**\r<BR>\n" );
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document.write( "* Var(Z) = Var(X + Y) = Var(X) + Var(Y) = 9 + 4 = 13<BR>\n" );
document.write( "* SD(Z) = sqrt(13) ≈ 3.61\r<BR>\n" );
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document.write( "**D) Z = X - Y**\r<BR>\n" );
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document.write( "* Var(Z) = Var(X - Y) = Var(X) + Var(Y) = 9 + 4 = 13<BR>\n" );
document.write( "* SD(Z) = sqrt(13) ≈ 3.61\r<BR>\n" );
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document.write( "**E) Z = -2X + 2Y**\r<BR>\n" );
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document.write( "* Var(Z) = Var(-2X + 2Y) = (-2)²Var(X) + 2²Var(Y) = 4 * 9 + 4 * 4 = 36 + 16 = 52<BR>\n" );
document.write( "* SD(Z) = sqrt(52) ≈ 7.21<BR>\n" );
document.write( " </SPAN>\n" );
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