8. Correct. The answer is true

(a) From Problem 2, we know that the mean of Y is m = E(Y)=1/2. Then, the variance is:

Var(Y)= E[(Y-m)2] = E[(Y-(1/2))2]

Var(Y)= ∫- (y - (1/2))2 f (y) dy

Var(Y)= ∫- (y - (1/2))2 (2e-2y ) dy = Var(Y) = 1/4

Alternatively, we can calculate the variance by using a theorem which establishes that:

Var[Y] = E[Y2] - (E[Y])2

Var(Y) = E[(Y-m)2]

Var(Y) = E(Y2) - [E(y)]2

Var(Y) = (1/2) - (1/2)2 =1/4

(b) s = [Var(Y)]0.5 = 1/2 (the standard deviation is equal to 500 barrels).