Inverse Laplace Transform to Recover CDF

Question

I know we can recover CDF of a variable random $ X $ by using inverse laplace transform. But as far as I know, laplace transform is defined on $ [0,\infty] $. Does it imply that we can only recover CDF of real positive random variable? How about bilateral laplace transform? Does inverse bilateral laplace transform can recover CDF of real variable random $ X $?

Answer 1

For the two-sided Laplace transform, one has to explicitly specify the region of convergence for the inverse transform. If the inverse transform of $F(p)$ is $f$ when $p$ lies in the vertical strip $ROC(f)$, then $$\mathcal L \!\left[ \int_{-\infty}^t f(\tau) d\tau \right] = \frac {F(p)} p, \\ p \in ROC(f) \land \operatorname{Re} p > 0.$$ This property can be used to obtain the cdf from the pdf $f$. If you take $p$ in the left half-plane, the answer will differ by $\int_{-\infty}^\infty f(\tau) d\tau = 1$.