Easy mathematics ?

[Maggette]

I can't contribute much to the formal side of the problem...

But you can play around a little bit in Matlab....and after a couple of "simulations"(1 000 000 random variables, different distributions...) I think it comes down to one problem...it is a good proxy as long x is small and constant, and that means small (for exsample 0.001) "on average" (read:expectation) AND "on average" constant (read: also a very small variance)...

edit: Sorry didn't want do bore you guys with a stupid post, but what I tried to say is, that you can't say if the proxy is "good" if you don't know what you need it for...So I think you need more information on the problem, like do I know which kind of distribution, do I know its moments etc.

[athletico]

>> Is E[ exp(x)] = exp( E[x] ) when x small (x stochastic)



I think I'm missing something in the way this is posed: if we only take the exponential out to first order then we'll certainly not account for the variance of x. Whether or not x is "small" is beside the point if x is stochastic.



To 2nd order:



E[ exp(x) ] = exp( E[x] ) * ( 1 + 1/2 Var(x) )

[Dimatrix]

Hmmm, a quick shot says no. Say x is N(mu,sigma) distributed, so normally. Then we have:



[img]/User%20Files/1559/Latex-Equation-7406.gif[/img]



But



[img]/User%20Files/1559/Latex-Equation-7407.gif[/img]



so



[img]/User%20Files/1559/Latex-Equation-7408.gif[/img]



So the two are different. No?

[pj]

Dimatrix

do not forget ≈

if σ is small it's Ok for outlier

[Dimatrix]

Well yes, if "small" for him is equal to "small variance" and if "approximately" is used instead of "equal to". But this then is rather trivial, as a random variable with a very small variance is almost deterministic.

[Dimatrix]

Ok, some more calculations seem to revise my last conclusion. If we have a uniformly distributed random variable



[img]/User%20Files/1559/Latex-Equation-7409.gif[/img]



Then we can calculate

[img]/User%20Files/1559/Latex-Equation-7410.gif[/img]



So, if the variance is very low, then a is close to b. But then



[img]/User%20Files/1559/Latex-Equation-7411.gif[/img]



So both are not the same even for very low variance. Yes?

[MadMax]

Dimatrix,



you missed that E[exp(x)]-> exp(b) or exp(a) since you have

[exp(b)-exp(a)]/(b-a) -> derivative(exp( b or a) ) when a->b

[Dimatrix]

This is true indeed. So, it's still valid that this variables are the same for low variances

[outlier]

Thanks every body for this clarifications. So the variance is the determinant of the validity of the relation [img]/User%20Files/4008/Latex-Equation-7414.gif[/img]. By saying x is small if it means that it almost always remains small say almost certainly between a and b. Than it is fine I guess because you get that the variance is less than (a-b)^2.

[crnt2]

Yes, the expression will be valid only for low variance.



The intuitive way to see this is that any variation of your random variable above the mean will have a large effect, because you are exponentiating. Any variation below the mean will have a much smaller effect. So for large variance you will always have



[img]/User%20Files/4031/Latex-Equation-7415.gif[/img].