Regarding the Laplace transform of a random variable (Basic Query)

Question

We know that the Laplace transform of a random variable is actually the Laplace transform of the probability density function of that random variable. Therefore my question is as follows. If we have a random variable $X$ then is it right to write the Laplace transform of $max(X,t)$ (here $t$ is a constant) as follows $$\int_0^{\infty} e^{-sx}f_X(x)dX$$ or will it be different? Many thanks in advance for the help.

Answer 1

If $X$ is a non-negative continuous random variable with density $f$ and $t>0$ is a constant then the Laplace Transform of $Y=\max\{X,t\}$ is: $$ \mathbb{E}[e^{-sY}]=\int_0^t e^{-st}f(x) dx+\int_t^\infty e^{-sx}f(x) dx=F(t)e^{-st}+\int_t^\infty e^{-sx}f(x) dx. $$