Question

Two random variables have zero mean and variances of 16 and 36. Their correlation coefficient is 0.5. Find:

a. Variance of their sum
b. Variance of their difference
c. Repeat (a) and (b) if the correlation coefficient is -0.5.

Answer 1

The variance of the sum and difference of two random variables remains the same (76 and 28, respectively) regardless of whether the correlation coefficient is 0.5 or -0.5.

Step-by-step explanation:

The task is to find the variance of the sum and difference of two random variables with zero mean and known variances, along with a given correlation coefficient. We use the properties of variance and the fact that for two random variables X and Y, the variance of their sum or difference is given by: Var(X ± Y) = Var(X) + Var(Y) ± 2Cov(X, Y), where Cov(X, Y) is the covariance of X and Y, which can be calculated using the formula Cov(X, Y) = ρ√Var(X)√Var(Y), with ρ being the correlation coefficient.

For a correlation coefficient of 0.5:

• a. Variance of their sum: Var(X+Y) = 16 + 36 + 2(0.5)√16√36 = 52 + 24 = 76.
• b. Variance of their difference: Var(X-​Y) = 16 + 36 - 2(0.5)√16√36 = 52 - 24 = 28.

For a correlation coefficient of -0.5:

• a. Variance of their sum: Var(X+Y) = 16 + 36 - 2(-0.5)√16√36 = 52 + 24 = 76.
• b. Variance of their difference: Var(X-​Y) = 16 + 36 + 2(-0.5)√16√36 = 52 - 24 = 28.

Note that the variance of their sum and difference remains the same regardless of whether the correlation coefficient is 0.5 or -0.5, because the covariance is squared in the variance formula which only affects the sign.