document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204179>Question 1204179</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given a random variable X, with standard deviation σX, and a random variable Y = a + bX, show<BR>\n" );
document.write( "that if b < 0, the correlation coefficient ρXY = −1, and<BR>\n" );
document.write( "if b > 0, ρXY = 1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847982 by </B><A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><B>ElectricPavlov(122)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=847982\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Find the Standard Deviation of Y**\r<BR>\n" );
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document.write( "* **Y = a + bX** <BR>\n" );
document.write( "* **σY = |b| * σX** <BR>\n" );
document.write( "    * The absolute value of 'b' is used because standard deviation must be non-negative.\r<BR>\n" );
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document.write( "**2. Find the Covariance of X and Y**\r<BR>\n" );
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document.write( "* **Cov(X, Y) = Cov(X, a + bX)** <BR>\n" );
document.write( "* **Cov(X, Y) = Cov(X, a) + Cov(X, bX)** <BR>\n" );
document.write( "* **Cov(X, Y) = 0 + b * Var(X)** <BR>\n" );
document.write( "    * Cov(X, a) = 0 because the covariance between a random variable and a constant is zero.<BR>\n" );
document.write( "    * Cov(X, bX) = b * Var(X) \r<BR>\n" );
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document.write( "* **Cov(X, Y) = b * σX²**\r<BR>\n" );
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document.write( "**3. Calculate the Correlation Coefficient (ρXY)**\r<BR>\n" );
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document.write( "* **ρXY = Cov(X, Y) / (σX * σY)** <BR>\n" );
document.write( "* **ρXY = (b * σX²) / (σX * |b| * σX)** <BR>\n" );
document.write( "* **ρXY = b / |b|**\r<BR>\n" );
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document.write( "**4. Determine the Sign of ρXY**\r<BR>\n" );
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document.write( "* **If b > 0:** <BR>\n" );
document.write( "    * ρXY = b / b = 1 <BR>\n" );
document.write( "    * This indicates a perfect positive linear relationship between X and Y.\r<BR>\n" );
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document.write( "* **If b < 0:**<BR>\n" );
document.write( "    * ρXY = b / (-b) = -1 <BR>\n" );
document.write( "    * This indicates a perfect negative linear relationship between X and Y.\r<BR>\n" );
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document.write( "**Therefore:**\r<BR>\n" );
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document.write( "* If b < 0, ρXY = -1<BR>\n" );
document.write( "* If b > 0, ρXY = 1\r<BR>\n" );
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document.write( "This demonstrates that the correlation coefficient between X and Y is perfectly positively or negatively correlated depending on the sign of the slope (b) in the linear relationship Y = a + bX.<BR>\n" );
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