CDF and MGF Relation

[S_David]

Hello,

Suppose that the Cumulative Distribution Function (CDF) of a random variable X is F_X(x), which is by definition is:

F_X(x)=\text{Pr}\left[X\leq x\right]=\text{Pr}\left[\frac{1}{X}\geq \frac{1}{x}\right]=1-\text{Pr}\left[\frac{1}{X}\leq \frac{1}{x}\right]=1-F_{1/X}\left(1/x\right)

Considering this relation between the CDF of X and the CDF of its reciprocal, what is the relation between the Moment Generating Function (MGF) of X and its reciprocal?

Any help will be highly appreciated.

Thanks in advance

[Pere Callahan]

A good starting point would be to think of a relation between the CDF of X and the MGF of X, wouldn't it?

[S_David]

A good starting point would be to think of a relation between the CDF of X and the MGF of X, wouldn't it?

Yes right, and I know what is the relation between them, but I want to see if another one has another idea. Anyway, the relation is:

M_X(s)=s\mathcal{L}\left\{F_X(x)\right\}

I have tried this, and it yields no where.

Regards

[bpet]

What would the CDF and MGF look like if X is uniform on [0,1] ?