Proving Bounds on Moments of a Random Variable

[silentone]

Homework Statement

For any random variable X, prove that
P{X$\geq$0}$\leq$inf[ E[ phi(t) : t $\geq$ 0] $\leq$ 1

where phi(t) = E[exp(tX)] o<phi(t)$\leq$∞

Homework Equations

The Attempt at a Solution

I am not sure how to begin this. Any hints to get started would be greatly appreciated.

 

[haruspex]

Homework Statement

For any random variable X, prove that
P{X$\geq$0}$\leq$inf[ E[ phi(t) : t $\geq$ 0] $\leq$ 1

E[ phi(t)] doesn't mean anything. E[] requires a r.v., whereas phi(t) is just an ordinary function. So I guess you mean
P{X$\geq$0}$\leq$inf[phi(t) : t $\geq$ 0] $\leq$ 1

Write out E[phi()] as an integral and consider the positive and negative ranges of X separately.

 

[Ray Vickson]

Homework Statement

For any random variable X, prove that
P{X$\geq$0}$\leq$inf[ E[ phi(t) : t $\geq$ 0] $\leq$ 1

where phi(t) = E[exp(tX)] o<phi(t)$\leq$∞

Homework Equations

The Attempt at a Solution

I am not sure how to begin this. Any hints to get started would be greatly appreciated.

Broad hint: P{X ≥ 0} = E H(X), where H(x) = 0 for x < 0 and H(x) = 1 for x ≥ 0. So, you are really comparing expectations of two different functions of X.

RGV