Question

Suppose the population standard deviation of X is 3 and the population standard deviation of Y is 5. Answer the following two questions, rounding to the nearest whole number (and remembering that variance is the square of standard deviation).

What is Var[2X - 8Y] if the covariance of X and Y is 2?


What is Var[2X - 8Y] if X and Y are independent?

Answer 1

The variance of 2X - 8Y with a covariance of 2 is 1572. If X and Y are independent, the variance of 2X - 8Y is 1636.

Step-by-step explanation:

To calculate the variance of a linear combination of random variables, use the formula:

Var[aX ± bY] = a² * Var[X] ± 2ab * Cov[X,Y] + b² * Var[Y]

1. With Covariance

Given that the population standard deviation of X is 3 and that of Y is 5, the variances are 9 and 25, respectively. With a covariance of 2 between X and Y, you compute the variance of 2X - 8Y:

Var[2X - 8Y] = (2² * Var[X]) - (2 * 2 * 8 * Cov[X,Y]) + (8² * Var[Y])

Var[2X - 8Y] = (4 * 9) - (32 * 2) + (64 * 25)

Var[2X - 8Y] = 36 - 64 + 1600

= 1572

2. With Independence

If X and Y are independent, the covariance is zero. So:

Var[2X - 8Y] = (2² * 9) + (8² * 25)

Var[2X - 8Y] = (4 * 9) + (64 * 25)

Var[2X - 8Y] = 36 + 1600

= 1636, rounded to the nearest whole number.