I have a problem with interpretation of one transformation performed on equation consisting of continuous random variables. Here is the source equation describing recurent relationship between the variables:
$Y_k = X_k + q(k,k-1) \cdot Y_{k-1}$
where:
$Y_k$ - random variable which describes time to find a fault while being in state k (k-faults already found)
$X_k$ - random variable which describes time spent in state k
$q(k,k-1)$ - probability of transition from state k to k-1, meanning that other fault is corrected before next fault is found
Knowing that pdf of $X_k$ is $f(t,k)$, and that pdf of $Y_k$ is $g(t,k)$, the equation is treated by the Laplace transform. The result is given as following:
$g^*(s,k) = f^*(s,k) \cdot g^*(q(k,k-1) \cdot s,k-1)$
My problem is to understand why the complex parameter "s" is multiplied by "$q(k,k-1)$" in this case. I would appreciate any help on this. Any theory explains such a result ?