Find the characteristic function of combination of random variables

Question

I have the same problem as here: Find characteristic function of ZX+(1-Z)Y with X uniform, Y Poisson and Z Bernoulli. Could you explain last equality? Can I get it without using law of total expectation?

Update

I have some idea, $E $$ e^{it(ZX + (1-Z)Y)} = E $$ (Ze^{itX} + (1-Z)e^{itY}) =$ $ = E $$ (Ze^{itX}) + E((1-Z)e^{itY}) = EZEe^{itX} + E(1-Z)Ee^{itY} = pEe^{itX} + (1-p)Ee^{itY}$

How to prove that $ e^{it(ZX + (1-Z)Y)} = Ze^{itX} + (1-Z)e^{itY} $ ?

Answer 1

Treat the cases $Z=0$ and $Z=1$ (the only two possible values of $Z$, since $Z$ has Bernoulli distribution).