document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1166252>Question 1166252</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Prove the symmetric property of Pearson�s correlation coefficient? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852861 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</B></A><img src=\"/images/stars/stars-13.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=852861\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The symmetric property of Pearson's correlation coefficient, $r_{X,Y}$, means that the correlation between variable $X$ and variable $Y$ is the same as the correlation between variable $Y$ and variable $X$.\r<BR>\n" );
document.write( "\n" );
document.write( "We need to prove that:<BR>\n" );
document.write( "$$r_{X,Y} = r_{Y,X}$$\r<BR>\n" );
document.write( "\n" );
document.write( "## 📐 Definition of Pearson's Correlation Coefficient\r<BR>\n" );
document.write( "\n" );
document.write( "Pearson's correlation coefficient, $r_{X,Y}$, is defined as the covariance of $X$ and $Y$ divided by the product of their standard deviations:\r<BR>\n" );
document.write( "\n" );
document.write( "$$r_{X,Y} = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}$$\r<BR>\n" );
document.write( "\n" );
document.write( "Where:<BR>\n" );
document.write( "* $\text{Cov}(X,Y)$ is the covariance between $X$ and $Y$.<BR>\n" );
document.write( "* $\sigma_X$ and $\sigma_Y$ are the standard deviations of $X$ and $Y$, respectively.\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "## 1. Covariance Property\r<BR>\n" );
document.write( "\n" );
document.write( "The **covariance** between two random variables $X$ and $Y$ is defined as:<BR>\n" );
document.write( "$$\text{Cov}(X,Y) = E[(X - \mu_X)(Y - \mu_Y)]$$\r<BR>\n" );
document.write( "\n" );
document.write( "The covariance between $Y$ and $X$ is defined as:<BR>\n" );
document.write( "$$\text{Cov}(Y,X) = E[(Y - \mu_Y)(X - \mu_X)]$$\r<BR>\n" );
document.write( "\n" );
document.write( "Since multiplication of real numbers is **commutative**, the order of the terms inside the expectation does not change the result:<BR>\n" );
document.write( "$$(X - \mu_X)(Y - \mu_Y) = (Y - \mu_Y)(X - \mu_X)$$\r<BR>\n" );
document.write( "\n" );
document.write( "Therefore, the covariance function is symmetric:<BR>\n" );
document.write( "$$\text{Cov}(X,Y) = \text{Cov}(Y,X)$$\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "## 2. Standard Deviation Property\r<BR>\n" );
document.write( "\n" );
document.write( "The standard deviation, $\sigma_X$, is an intrinsic property of the single variable $X$, and $\sigma_Y$ is an intrinsic property of $Y$.\r<BR>\n" );
document.write( "\n" );
document.write( "The product of the standard deviations is also commutative:<BR>\n" );
document.write( "$$\sigma_X \sigma_Y = \sigma_Y \sigma_X$$\r<BR>\n" );
document.write( "\n" );
document.write( "---\r<BR>\n" );
document.write( "\n" );
document.write( "## 3. Proof of Symmetry\r<BR>\n" );
document.write( "\n" );
document.write( "Using the definition of the correlation coefficient, we can write the formula for $r_{Y,X}$:\r<BR>\n" );
document.write( "\n" );
document.write( "$$r_{Y,X} = \frac{\text{Cov}(Y,X)}{\sigma_Y \sigma_X}$$\r<BR>\n" );
document.write( "\n" );
document.write( "Now, substitute the symmetric properties established in Step 1 and Step 2 into the expression for $r_{Y,X}$:\r<BR>\n" );
document.write( "\n" );
document.write( "$$r_{Y,X} = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}$$\r<BR>\n" );
document.write( "\n" );
document.write( "Since the right side of this equation is the definition of $r_{X,Y}$:<BR>\n" );
document.write( "$$r_{Y,X} = r_{X,Y}$$\r<BR>\n" );
document.write( "\n" );
document.write( "Thus, Pearson's correlation coefficient possesses the **symmetric property**. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );