Suppose a random variable $B$ is constructed from three random variables $X, Y, Z$ as follows $B=X+Y+Z$ with $X, Y$ constant and equal to $1$ and $Z \sim exp(1)$.
I am confused which of the two Laplace-Stieljens transforms stated below is correct.
option 1: We can write $B = X+Y+Z = 2+Z$. Then the using $\tilde B(s)=\tilde (X+Y)(s) * \tilde Z(s)=\frac{2}{s}*\frac{1}{1+s}$
option 2: $\tilde B(s)=\tilde X(s)*\tilde Y(s) * \tilde Z(s)=\frac{1}{s}*\frac{1}{s}*\frac{1}{1+s}$
The random variable $B$ is actually the total processing time of a certain job which consists of three phases of which two take a constant time of $1$ hour and the processing time of the third job is exponentially distributed.