Fall 2000 exam #17 (bad question? or am I missing something?

[Nastasya]

Hi guys,

I'm working on this problem and have run into a bit of trouble. Here is the problem statement. Please bear with me, I really need help here.

A stock market analyst has recorded the daily sales revenue for two companies over the last yr and displayed them in the histograms below.

picture of histogram for company A

picture of histogram for comany B

since I cant draw the pictures, histogram A and B appear to have the same mean (100) with histogram B being more spread out (larger standard deviation).

The analyst noticed that a daily sales revenue above 100 for Company A was always accompanied by a daily sales revenue below 100 for Company B and vice versa.

Let X denote the daily sales revenue for Company A and let Y denote the daily sales revenue for Company B on some future day.

Assuming that for each company the daily sales revenues are independent and identically distributed, which of the following is true?

A) Var X > Var Y and Var (X+Y) > Var(X)+Var(Y)
B) Var X > Var Y and Var (X+Y) < Var (X) + Var (Y)
C) Var X > Var Y and Var (X+Y) = Var (X) + Var (Y)
D) Var X < Var Y and Var (X+Y) > Var (X) + Var (Y)
E) Var X < Var Y and Var (X+Y) < Var (X) + Var (Y)

Ok so clearly Var X < Var Y since Company A has a smaller standard deviation then Company B. Here the solution and I agree. But then
Var (X+Y) = Var X + Var Y since we're assuming that they are independent. Covariance is 0 when two rvs are independent!!! In the solution they have that Var(X+Y)= Var X + Var Y + 2 Cov (X,Y). What am I missing here?

Any input is greately appreciated

ps their answer is E?