I calculated the the moment generating function of an random variable $W$ and got the following: $$M_W(s)=(1-\rho)+ \rho \frac{\mu(1-\rho)}{\mu(1-\rho)-s},$$ where $\mu$ and $\rho$ are two parameters.
Let $X$ be an exponential variable with parameter $\mu(1-\rho)$. Hence the moment generating function is given by: $$M_X(s)=\frac{\mu(1-\rho)}{\mu(1-\rho)-s}.$$ And I also know that $M_X(0)$ is always $1$ (if MGF exits). So I have the following: $$M_W(s)=(1-\rho)M_X(0)+ \rho M_X(s).$$ And from this I can conclude that $W$ is with probability $(1-\rho)$ equal to zero and with probability $\rho$ equal to an exponential variable with parameter $\mu(1-\rho)$? Can somebody explain it to me?
And how can I get from that the distribution function $\Pr(W \leq t)$?