This is the question:
A and B are two competing companies. An investor decides whether to buy
(a) 100 shares of A, or
(b) 100 shares of B, or
(c) 50 shares of A and 50 shares of B.
A profit made on 1 share of A is a random variable X with the distribution P (X = 2) = P (X = −2) = 0.5.
A profit made on 1 share of B is a random variable Y with the distribution P(Y = 4) = 0.2,P(Y = −1) = 0.8.
If X and Y are independent, compute the expected value and variance of the total profit for strategies (a), (b), and (c).
Here is what I have:
E(aX) = aE(X) and Var(aX) = a^2Var(X)
a) E(X) = .5*2 + .5*-2 = 0 so aE(X) = 0
Var(X) = 100^2((.5)(2-0)^2 + (.5)(-2-0)^2) = 40,000
b) E(Y) = 4*.2 + 1*.8 = 0 so aE(Y) = 0
Var(Y) = 100^2((.2)(4-0)^2 + (.8)(-1-0)^2) = 40,000
c) (independent) so E(aX + bY) = aE(X) + bE(Y) = 0
independent so Var(aX + bY) = a^2Var(X) + b^2Var(Y)
50^2((.2)(4-0)^2 + (.8)(-1-0)^2) = 10,000 50^2((.5)(2-0)^2 + (.5)(-2-0)^2) = 10,000
so Var(aX + bY) = 20,000
I'm asking this question because these values seem way off. I'm not sure what I'm doing wrong as I am going by definitions and properties outlines in the textbook.