What is the distribution of the product between a continuous (Exponential) and a discrete (Bernoulli) random variables?

Question

If you have $X$ distributed as a $\mathrm{Bernoulli}(p)$ and $Y$ as a $\mathrm{Exponential}(\lambda)$ find $Z=XY$.

I tried doing it with the MFG i.e. $$E[e^{tZ}]=E[e^{tXY}]=E[E[e^{tXY}|_{X=x}]=E[E[e^{txY}]|_{X=x}]$$ but this is where I got stuck, I don´t know if this is the proper way to solve this, I would appreciate a step-by-step explanation.

Answer 1

$E[e^{tXY}]=(1-p)E[e^{tY \times 0}] + pE[e^{tY \times 1}] =(1-p)E[1] + pE[e^{tY}]$. Does that help?