Variance of Sum and Difference for Correlated Data

This solution explains how to find the value of $\text{Var}(X+Y)+\text{Var}(X-Y)$ when dealing with two correlated data series, X and Y.

Key Variance Formulas Explained

To solve this problem, we utilize the standard formulas for the variance of the sum and difference of two random variables (or data series). These formulas take into account whether the variables are correlated.

• Variance of the Sum: The variance of $X+Y$ is given by:

  $ \text{Var}(X+Y) = \text{Var}(X) + \text{Var}(Y) + 2\text{Cov}(X, Y) $

• Variance of the Difference: The variance of $X-Y$ is given by:

  $ \text{Var}(X-Y) = \text{Var}(X) + \text{Var}(Y) - 2\text{Cov}(X, Y) $

In these formulas:

• $\text{Var}(X)$ is the variance of data series X.
• $\text{Var}(Y)$ is the variance of data series Y.
• $\text{Cov}(X, Y)$ is the covariance between X and Y. This term captures how the two data series move together. If X and Y are correlated, this term is non-zero.

Step-by-Step Calculation

We are asked to find the sum $\text{Var}(X+Y)+\text{Var}(X-Y)$. Let's substitute the formulas above:

$ \text{Var}(X+Y) + \text{Var}(X-Y) = \left( \text{Var}(X) + \text{Var}(Y) + 2\text{Cov}(X, Y) \right) + \left( \text{Var}(X) + \text{Var}(Y) - 2\text{Cov}(X, Y) \right) $

Now, we simplify the expression by combining like terms:

1. Combine the $\text{Var}(X)$ terms: $\text{Var}(X) + \text{Var}(X) = 2\text{Var}(X)$
2. Combine the $\text{Var}(Y)$ terms: $\text{Var}(Y) + \text{Var}(Y) = 2\text{Var}(Y)$
3. Combine the covariance terms: $2\text{Cov}(X, Y) - 2\text{Cov}(X, Y) = 0$

Adding these combined terms gives us:

$ 2\text{Var}(X) + 2\text{Var}(Y) + 0 $

This simplifies to:

$ 2\text{Var}(X) + 2\text{Var}(Y) $

We can factor out the common factor of 2:

$ 2(\text{Var}(X) + \text{Var}(Y)) $

Final Result

The calculation shows that $\text{Var}(X+Y)+\text{Var}(X-Y)$ simplifies to $2(\text{Var}(X)+\text{Var}(Y))$. Notice how the covariance term, which represents the correlation, cancels out in this specific sum.