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?

[Gandalf]

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?

Phrase highlighted in red says (at least in the past) the covariance is negative.

Does "independent and identically distributed" contradict that? With some misgivings, I would think not. Surely in this problem you are not supposed to think that X and Y are identically distributed, so you likewise should not think X and Y are independent and identically distributed.

Given the phrase highlighted in blue, the intent is clearly that Company A's sales on any given day are independent and identically distributed as Company A's sales on any other day; likewise for Company B.

That is clearly where SOA is coming from. Wording may not be a thing of beauty.

[Nastasya]

Aha! Thanks for clearing this up for me. :P