Does E(e^{-X}) = 0 imply X = \infty almost surely for X \geq 0?

[osprey]

Does the following make sense:

$<br /> E(e^{-X}) = 0 \Rightarrow X = \infty\quad a.s. ?<br />$

(Intuitively yes, but mathematically?)

Thank you in advance for your help! :-)

/O

 

[micromass]

Hi osprey!

Yes, that makes sense! Can you show in general for $X\geq 0$ that

$$E[X]=0~\Rightarrow~X=0~\text{a.s.}$$

Then you just need to apply this result for $e^{-X}$...