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).