MGF of the product of a exponential and a bernoulli random variable Let $=$ , where $X$ and $Y$ are independent,  ~(0.01) and ∼(0.3)
Is there a way to find the m.g.f of ?
I know that I can find the C.D.F by doing as explained here.
Initially I tried to find de P.D.F using the C.D.F and determine the M.G.F by hand, but I can't find a way to determine the P.D.F too.

By using law of total expectation:
$$\mathbb{E}[e^{tXY}] = \mathbb{E}[\mathbb{E}[e^{tXY}|Y] = \mathbb{E}[e^{tXY}|Y=0]P(Y=0) + \mathbb{E}[e^{tXY}|Y=1]P(Y=1) = \mathbb{E}[e^{0}]0.7 + \mathbb{E}[e^{tX}]0.3 = 0.7 + 0.3M_{X}(t)$$
