Question

Let 1, 2, 3, and 4 be independent random variables with a mean of 10 and a variance of 16. Suppose that \( Y = 2X_1 - 3X_2 + X_3 + 0.5X_4 \). What is the mean and variance of \( Y \)?

A) Mean = 10, Variance = 16
B) Mean = 5, Variance = 8
C) Mean = 20, Variance = 32
D) Mean = 0, Variance = 4

Answer 1

The mean of Y is 5 and the variance of Y is 228.

Step-by-step explanation:

To find the mean and variance of Y, we need to calculate the mean and variance of each term in the expression for Y and then add them up.

Given that the mean of each random variable is 10 and the variance is 16, we have:

• The mean of 2X1 is 2 times the mean of X1, which is 2*10 = 20.
• The mean of -3X2 is -3 times the mean of X2, which is -3*10 = -30.
• The mean of X3 is the mean of X3, which is 10.
• The mean of 0.5X4 is 0.5 times the mean of X4, which is 0.5*10 = 5.

So, the mean of Y is 20 - 30 + 10 + 5 = 5.

To calculate the variance of Y, we use the fact that if X and Y are independent random variables, then the variance of a linear combination aX + bY is a^2 times the variance of X plus b^2 times the variance of Y. Applying this formula to Y, we have:

• The variance of 2X1 is (2^2)*(16) = 4*16 = 64.
• The variance of -3X2 is (-3^2)*(16) = 9*16 = 144.
• The variance of X3 is 16.
• The variance of 0.5X4 is (0.5^2)*(16) = 0.25*16 = 4.

So, the variance of Y is 64 + 144 + 16 + 4 = 228.