Derivation of the expectation of an exponential

Question

I would like to demonstrate the formula which states that if

$X$ is a random variable such that

$\forall \lambda >0, E[e^{-\lambda^2.\frac{X}{2}}] = e^{-\lambda.b}$

Where $b$ is a positive constant.

Then by derivation regardng $\lambda^2$ and taking $\lambda = 0$.

Once can calculate $E[X]$

In order to demonstrate this statement, I thought of using whole series to demonstrate such a purpose. However this lead me nowhere...

Can you please advise?!!

Thank you

Answer 1

The hypothesis can be written as $Ee^{-tX}=e^{-\sqrt {2t} b}$ for all $t>0$. Differentiating w.r.t. $t$ (from the right) and setting $t=0$ we get $EX=\infty$.