Question 1191144
Here's how to calculate the mean (expected value) of Z for each case, given E(X) = 20 and E(Y) = 40:

**Key Principle:**

E(aX + bY + c) = aE(X) + bE(Y) + c, where a, b, and c are constants.

**A) Z = 25 - (1/2)X**

E(Z) = E(25 - (1/2)X)
E(Z) = 25 - (1/2)E(X)
E(Z) = 25 - (1/2)(20)
E(Z) = 25 - 10
E(Z) = 15

**B) Z = (1/3)X - 8**

E(Z) = E((1/3)X - 8)
E(Z) = (1/3)E(X) - 8
E(Z) = (1/3)(20) - 8
E(Z) = 20/3 - 24/3
E(Z) = -4/3  (approximately -1.33)

**C) Z = X + Y**

E(Z) = E(X + Y)
E(Z) = E(X) + E(Y)
E(Z) = 20 + 40
E(Z) = 60

**D) Z = X - Y**

E(Z) = E(X - Y)
E(Z) = E(X) - E(Y)
E(Z) = 20 - 40
E(Z) = -20

**E) Z = -3X + 3Y**

E(Z) = E(-3X + 3Y)
E(Z) = -3E(X) + 3E(Y)
E(Z) = -3(20) + 3(40)
E(Z) = -60 + 120
E(Z) = 60