Question 1166252
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**.