Posted on September 21 2006

Who can help us out?

The following messeage is sent by LittleZ on Thursday night...
Xiaoxue says:
one question for you
Xiaoxue says:
we knew x>0, a>0,b>0
Xiaoxue says:
under what condition(i mean an expression about a and b),
Xiaoxue says:
we will have:
Xiaoxue says:
(Mx(a))^(1/a) larger than (Mx(b) )^(1/b)?
Xiaoxue says:
where Mx(a) and Mx(b) are the moment generating functions of x.
Xiaoxue says:
help me if you can carol, xiaocao and i are stuck in the office now.
Xiaoxue says:
we wanna go home but the problem is due tomorrow morning~
Xiaoxue says:
Xiaoxue says:
just get one expression contains a and b (no matter it is equality or inequality)
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Latest Updates @ 12:30 Sept 22 2006 Fri:
We ask the prof to extend the due date(next monday). here are some directions we might go with and work on:
(1) Take x as a certain distribution, view a and b as variables of the moment generating functions. Take derivative both on a and b, figure out Mx(c) isincreasing or decreasing(my intuition tells me it is an increasing function tho....). Then draw about graph with a/b on the x-axis, Mx( ) on the y-axis -------------hopefully we would be able to draw some kinda result from it.
(2) Try to use Taylor expansion on Mx(a), simply we could view Mx(a) as the expectation of e^(ax), Christine showed her deduction this morning, and got a>b. I will go with the first method, and try to verify her answer.
(3) Okay, this one is really funny. i got this idea on the bus this morning. Because this question is originally from a Utility Context, i would like to borrow the idea of indifference curve in Economics. I supposed the indifference curves are all taken with similar pattern(parallel), in order to have A>B, the curve A must always higher than Curve B at every point of x and keeping the slope at every tangent point equal(otherwise, the two will intercept somewhere, sometime). So we may take the derivative on both sides of the generating functions, let n=0, get the first moment which is the mean/slope, and get an expression of a and b.
BUT, I failed, since I don't know how to deal with the powers of (1/a) and (1/b) .
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I will keep trying this prob. this weekend. and get you guys some feedback(err... is there really anybody here? or i'm just talking to myself...)