Understanding Correlation and Uncorrelated Variables

This question asks us to find the correlation between two new variables, \(X+Y\) and \(Y+Z\), given that \(X\), \(Y\), and \(Z\) are three uncorrelated variables with variances \(\sigma_x^2\), \(\sigma_y^2\), and \(\sigma_z^2\) respectively.

When variables are uncorrelated, it means their covariance is zero. So, we know that:

• \(\text{Cov}(X, Y) = 0\)
• \(\text{Cov}(X, Z) = 0\)
• \(\text{Cov}(Y, Z) = 0\)

The correlation coefficient between two variables, say A and B, is defined as:

$$\rho(A, B) = \frac{\text{Cov}(A, B)}{\sqrt{\text{Var}(A)\text{Var}(B)}}$$

We need to calculate the covariance between \(X+Y\) and \(Y+Z\), and the variances of \(X+Y\) and \(Y+Z\).

Calculating Covariance(\(X+Y\), \(Y+Z\))

Using the properties of covariance (linearity):

$$\text{Cov}(X+Y, Y+Z) = \text{Cov}(X, Y+Z) + \text{Cov}(Y, Y+Z)$$

Let's break down each term:

• \(\text{Cov}(X, Y+Z) = \text{Cov}(X, Y) + \text{Cov}(X, Z)\). Since \(X, Y, Z\) are uncorrelated, \(\text{Cov}(X, Y) = 0\) and \(\text{Cov}(X, Z) = 0\). So, \(\text{Cov}(X, Y+Z) = 0 + 0 = 0\).
• \(\text{Cov}(Y, Y+Z) = \text{Cov}(Y, Y) + \text{Cov}(Y, Z)\). We know \(\text{Cov}(Y, Y) = \text{Var}(Y) = \sigma_y^2\). Since \(Y\) and \(Z\) are uncorrelated, \(\text{Cov}(Y, Z) = 0\). So, \(\text{Cov}(Y, Y+Z) = \sigma_y^2 + 0 = \sigma_y^2\).

Therefore, the covariance between \(X+Y\) and \(Y+Z\) is:

$$\text{Cov}(X+Y, Y+Z) = 0 + \sigma_y^2 = \sigma_y^2$$

Calculating Variance(\(X+Y\)) and Variance(\(Y+Z\))

Using the properties of variance for uncorrelated variables:

• \(\text{Var}(X+Y) = \text{Var}(X) + \text{Var}(Y)\) because \(X\) and \(Y\) are uncorrelated. So, \(\text{Var}(X+Y) = \sigma_x^2 + \sigma_y^2\).
• \(\text{Var}(Y+Z) = \text{Var}(Y) + \text{Var}(Z)\) because \(Y\) and \(Z\) are uncorrelated. So, \(\text{Var}(Y+Z) = \sigma_y^2 + \sigma_z^2\).

Calculating the Correlation Coefficient

Now, we can plug these values into the correlation formula:

$$\rho(X+Y, Y+Z) = \frac{\text{Cov}(X+Y, Y+Z)}{\sqrt{\text{Var}(X+Y)\text{Var}(Y+Z)}}$$

$$\rho(X+Y, Y+Z) = \frac{\sigma_y^2}{\sqrt{(\sigma_x^2 + \sigma_y^2)(\sigma_y^2 + \sigma_z^2)}}$$

The correlation between \(X+Y\) and \(Y+Z\) is \(\frac{\sigma_y^2}{\sqrt{(\sigma_x^2 + \sigma_y^2)(\sigma_y^2 + \sigma_z^2)}}\). This value depends on the specific variances \(\sigma_x^2\), \(\sigma_y^2\), and \(\sigma_z^2\).

Evaluating the Options

Let's consider the given options:

• Option 1: 1/2 - This would only be true under specific conditions, e.g., if \(\sigma_x^2 = \sigma_y^2 = \sigma_z^2\). In that case, the correlation becomes \(\frac{\sigma_y^2}{\sqrt{(2\sigma_y^2)(2\sigma_y^2)}} = \frac{\sigma_y^2}{\sqrt{4\sigma_y^4}} = \frac{\sigma_y^2}{2\sigma_y^2} = 1/2\) (assuming \(\sigma_y^2 > 0\)).
• Option 2: 1 - This would require the numerator to equal the denominator (and the denominator to be positive). This happens only under very specific degenerate conditions where the relationship is perfectly linear, which is not guaranteed by just being uncorrelated with a third variable. For example, if \(\sigma_x^2=0\) and \(\sigma_z^2=0\), then \(\rho = \frac{\sigma_y^2}{\sqrt{(\sigma_y^2)(\sigma_y^2)}} = 1\), but this implies X and Z are constant variables.
• Option 4: 0 - This would require the covariance to be zero, which means \(\sigma_y^2 = 0\). This implies Y is a constant variable.

Since the question gives us general uncorrelated variables with unspecified variances, the correlation coefficient is generally given by the derived formula \(\frac{\sigma_y^2}{\sqrt{(\sigma_x^2 + \sigma_y^2)(\sigma_y^2 + \sigma_z^2)}}\). This value is not necessarily 1/2, 1, or 0 unless specific conditions on the variances are met. Therefore, based on the general information provided, the correlation is none of these fixed values.

Conclusion

The correlation between \(X+Y\) and \(Y+Z\) for uncorrelated variables \(X, Y, Z\) with variances \(\sigma_x^2, \sigma_y^2, \sigma_z^2\) is \(\frac{\sigma_y^2}{\sqrt{(\sigma_x^2 + \sigma_y^2)(\sigma_y^2 + \sigma_z^2)}}\). This expression shows that the correlation depends on the individual variances and is not a fixed value like 1/2, 1, or 0 in the general case. Hence, the correct option is "none of these".

Revision Table: Key Concepts

Additional Information: Uncorrelated vs Independent

It's important to distinguish between uncorrelated and independent variables.

• Independence: If two random variables X and Y are independent, then knowing the value of one variable provides no information about the value of the other. Independence is a strong condition related to the joint probability distribution.
• Uncorrelated: If two random variables X and Y are uncorrelated, their covariance \(\text{Cov}(X, Y)\) is 0. This only measures the absence of a linear relationship.

If two variables are independent, they are always uncorrelated. However, the reverse is not always true. Variables can be uncorrelated but not independent, especially if the relationship between them is non-linear (e.g., \(Y = X^2\) where X is symmetrically distributed around 0). In this question, we are only told the variables are uncorrelated, which is sufficient for the variance and covariance calculations used above.