Another Variance and Expectation of diffrent portfolios

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).

--- For E(X) for both A and B I get: EA(X) =(2)(.5) + (-2)(.5) = 0. EB(X) =(4)(.2) + (-1)(.8) = 0.

To obtain variance: EA(X^2)= (2^2)(.5) + (-2^2)(.5) = 0. EB(X^2)= (4^2)(.2) + (-1^2)(.8) = 3.2 + .8 = 4

VarA(X) = EA(x^2) - EA(X)^2 = 0 - 0^2 = 0 VarB(X) = EB(x^2) - EB(X)^2 = 4 - 0^2 = 4

a) 100 shares * X profit = 100X = A

E(A) = E(100x) = 100E(x) = 100 * 0 = 0 Var(A) = 100^2*Var(X) = 10,000 * 1 = 0

b) 100 shares * X profit = 100X = B

E(B) = E(100x) = 100E(x) = 100 * 0 = 0 Var(B) = 100^2*Var(X) = 10,000 * 4 = 40,000

c) 50 shares * X profit + 50 shares * X profit = 50X + 50Y = Z

E(Z) = EA(50x) + EB(50X)= 50EA(X) + 50EB(X)= 50*0 + 50* 0 = 0 Var(Z) = 50^2*VarA(X) + 50^2*VarB(X) =2500*0 + 2500 * 4 = 10,000

That one I do not have a clue if my answers are right or wrong but, I am really doubting myself. Can any one confirm or correct me? Thank you!

Answer 1

Well, you have a few things right and a few things wrong.
Your basic calculations for mean and variance are correct.
Also, your understanding of the probability of a random variable in a discrete distribution.

However, you cannot say that E(100X) = 100E(X) or that V(100X) = 100^2V(X)
Every share traded is a separate random variable!
Actually, denoting 100X is actually saying that you are multiplying the "Probability" of the random variable by a constant, which is something completely different, and V(100X) = 100^2V(X) would actually be correct! but this is not the case here.

If we were to trade two shares of A and wanted to calculare the mean and variance we would have to write it as E(X + Y) = E(X) + E(Y) where in this case X = Y.
Same goes for variance: V(X + Y) = V(X) + V(Y)

In your example V(B) is not 40000 but actually 400.

I advise you to look at this page from Yale university: Mean and Variance of a Random variable
Also, look at this example of 2 dice being thrown (this would give you an intuition to looking at each share traded as a dice being thrown): Two dice thrown - Mean and Variance

All the best!