I have an expression in the from:
$$\mathbb{P}(H>\theta)=\int_0^{\infty}\exp(-m\theta I)f_I(i){\rm{d}}i$$
Here, $f_I(i)$ is the PDF of random variable $I$.
Let, $\mathcal{L}_I(s)$ is the Laplace transform of $f_I(i)$,
How can we express $\mathbb{P}(H>\theta)$ in terms of $\mathcal{L}_I(s)$ when the Laplace transform of $I$ is evaluated at $s=m\theta using the properties of Laplace transform?
Conclusion: I want to express the right hand side of the equation in terms of Laplace transform $\mathcal{L}_I(s)$, wh.ere $\mathcal{L}_I(s)$ is the Laplace transform of $f_I(i)$.