# statistics (expected value, variance)

• Oct 10th 2012, 04:59 PM
dumbledore
statistics (expected value, variance)
Air Canada and United have for many years been members of the star alliance, selling seats on each others flights while competing for passengers. In 2011 they proposed closer collaboration. The Competition bureau blocked their proposition on the grounds it was uncompetitive. Air Canada rejected the competition bureaus decision and went before the competition tribunal for a decision. Since this was an unusual case, it was tough to predict the outcome and some lawyers put it at 50/50.

Suppose the following table represents the increase to Air Canadas profits under "good" or "bad" economic conditions which have a probability of 0.3 and 0.7, respectively.

Economic conditions

Tribunal outcome: Win +840m +210m
Lose +150m -500m

i) what is the expected increase in profits to Air Canada?
ii) what is the variance of the increase in profits to Air Canada?
iii) how do answer to i) and ii) compare to just accepting the competition bureaus decision?
iv) is it worth Air Canada spending 32 m on lawyer fees to fight the case?

for i) I used Expected value = sum x * P(x) =
E(x)= 840,000,000 (0.3) + 210,000,000 (0.7) = 399,000,000. I don't know how to do the rest or if I'm on the right track
• Oct 10th 2012, 07:31 PM
chiro
Re: statistics (expected value, variance)
Hey dumbledore.

In this problem you have a joint distribution where one variable is Win/Lose and the other is Good/Bad in terms of economic conditions. We assume that each variable is independent which gives the joint distribution P(X = Win/Lose, Y = Good/Bad) = P(X=Win/Lose)*P(Y=Good/Bad).

So for the expected profits, we need to take into account the total joint distribution which has four possibilities (unless specified otherwise, but your question has not).

So the expectation is E[E[X|Y]] = 840,000,000*(0.5)*(0.3) + 210,000,000*(0.7)*(0.5) + 150,000,000*(0.5)*(0.3) - 500,000,000*(0.3)*(0.5) = 147,000,000 based on the calculation in R:

> 840*0.3*0.5 + 210*0.7*0.5 + 150*0.3*0.5 - 500*0.3*0.5
[1] 147

Remember that you need to take into account all possibilities not just the situation where you win.

The situation where you win is basically E[Expected Winnings|You win the tribunal] which is what you were looking at but you can't assume you won unless specifically informed, and it says that there was no outcome yet and that the prediction was 50/50.

Other expectations include things like E[Expected Winnings|Bad economic conditions] as an example.
• Oct 12th 2012, 12:06 PM
dumbledore
Re: statistics (expected value, variance)
But when I try to compute the variance using Var(x)=(x-E(x))^2 * P(x) from the expected value of 47,000,000 (you miscalculated) I get a ridiculously large number. How do I compute the variance? I did it by using the formula: Var(x)=(840m-47m)^2 * (0.3)(0.5) + ..... etc
• Oct 12th 2012, 04:55 PM
chiro
Re: statistics (expected value, variance)
Can you show us your calculation for the variance? Also looking at the data, I would expect the variance to be quite large.