MGF and the Normal Distn

Hi,

I am struggling with Que 6.30 and exam-type question which follows. In Summary it shows that 2X-3Y~N(2mux-3muy,4sigma2x+9sigma2y) However, (X1+X2)-(Y1+Y2+Y3)~N(2mux-3muy,2sigma2x+3sigma2y) (excuse the notation - haven't worked out how to easily add Greek symbols). I can't see why the 2 distributions are different or more to the point what is a practical example of the 2 which highlights the differences.
Thanks,
Stewart.

 

The point is basically that if you have something of the form aX, any deviations from the norm will be magnified by a factor of a, so you expect a higher variance (in fact, you get var(aX)=a^2*var(X) ). If however, you've got two identical, independent distributions (call them X, Y) then X+Y will not magnify any errors in the same way - due to independence you get var(X+Y)=var(X)+var(Y)=2var(X) - that is the variations add up, rather than get multiplied.
That was sort of just a mind-dump, don't know if any of that helped at all?

 

That does help - but whats the difference (in some real life situation) between 2X and X1 + X2?

 

Simple example of gambling $2 on heads with 1 coin toss or $1 each on 2 tosses.

Payout on first is (-2 ,+2) each with probability 0.5.
Payout on second is (-2,0,+2) with probability (0.25,0.5,0.25). You should see that this has lower variance.

 

Thanks - those 2 examples are helpful. So basically if we have multiple events we are looking at a model like X1+X2 whereas a single event doubled in some way would look more like 2X? So with the Exam-type question for Chapter 3 where X is home insurance claims ~N(800,10,000), several claims give rise to a X1+X2 type model, what would give rise to a model that looked like 2X?

 

2X would be where we simply took one measurement and doubled it. For a contrived example, suppose an insurer has two policies in force, and decides to use the next claim as data for the overall set. Since there will only be one claim, they'll have to double it to get a 'X+Y' style thing, getting 2X. Obviously this would be a rather foolish insurer (as you've seen before, any variations get amplified here, rather than smoothed off a bit, as would happen for X+Y), but that would give the kinda thing you're looking for I think?

 

That all makes sense - thanks all.