Question 1166252: Prove the symmetric property of Pearson�s correlation coefficient?
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website! 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 $X$.
We need to prove that:
$$r_{X,Y} = r_{Y,X}$$
## 📐 Definition of Pearson's Correlation Coefficient
Pearson's correlation coefficient, $r_{X,Y}$, is defined as the covariance of $X$ and $Y$ divided by the product of their standard deviations:
$$r_{X,Y} = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}$$
Where:
* $\text{Cov}(X,Y)$ is the covariance between $X$ and $Y$.
* $\sigma_X$ and $\sigma_Y$ are the standard deviations of $X$ and $Y$, respectively.
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## 1. Covariance Property
The **covariance** between two random variables $X$ and $Y$ is defined as:
$$\text{Cov}(X,Y) = E[(X - \mu_X)(Y - \mu_Y)]$$
The covariance between $Y$ and $X$ is defined as:
$$\text{Cov}(Y,X) = E[(Y - \mu_Y)(X - \mu_X)]$$
Since multiplication of real numbers is **commutative**, the order of the terms inside the expectation does not change the result:
$$(X - \mu_X)(Y - \mu_Y) = (Y - \mu_Y)(X - \mu_X)$$
Therefore, the covariance function is symmetric:
$$\text{Cov}(X,Y) = \text{Cov}(Y,X)$$
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## 2. Standard Deviation Property
The standard deviation, $\sigma_X$, is an intrinsic property of the single variable $X$, and $\sigma_Y$ is an intrinsic property of $Y$.
The product of the standard deviations is also commutative:
$$\sigma_X \sigma_Y = \sigma_Y \sigma_X$$
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## 3. Proof of Symmetry
Using the definition of the correlation coefficient, we can write the formula for $r_{Y,X}$:
$$r_{Y,X} = \frac{\text{Cov}(Y,X)}{\sigma_Y \sigma_X}$$
Now, substitute the symmetric properties established in Step 1 and Step 2 into the expression for $r_{Y,X}$:
$$r_{Y,X} = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}$$
Since the right side of this equation is the definition of $r_{X,Y}$:
$$r_{Y,X} = r_{X,Y}$$
Thus, Pearson's correlation coefficient possesses the **symmetric property**.
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