If $ E X \gt 0$, is $G'(0) \gt 0 $?

Question

Let X be a random variable, let $G_X(t)$ be its characteristic function. If $ E X \gt 0$, is $G'(0) \gt 0 $ ?

By definition, $G_X'(t)= iE( Xe^{itX})$ so $G_X'(0)= iE( X)$

Since $G_X'(0)$ is a complex number, how can I determine if it is always positive?

Also, I'm wondering if $ E X = \infty$, is $G_X'(0)$ defined?

Answer 1

If $X$ has Cauchy distribution then $G_X (t) =e^{-|t|}$ for which $G_X'(0)$ does not exist. ($EX=\infty$ in this case). This answers the second part. For the first part your argument is correct; it is wrong to say that $EX>0$ implies $G_X'(0)>0$.