Find characteristic function of ZX+(1-Z)Y with X uniform, Y Poisson and Z Bernoulli

Question

Let random variables $X, Y, Z$ independent. $X$ with uniform distribution on $[-a,a]$, $Y$ with Poisson distribution with parameter $ν$, $Z$ with Bernoulli distribution with parameter $p$.

Find the characteristic function of random variable $Z*X+(1-Z)Y$

I tried use the this idea, but i don't now, how i can apply it for discrete distribution.

Why are they independent?(random variables)

No, i can't understand the conversion to replacement Z by p and (1-p). Why we can do it?It is not the meaning of characteristic function.

Answer 1

$$T=ZX+(1-Z)Y\implies E(\mathrm e^{\mathrm itT})=pE(\mathrm e^{\mathrm itX})+(1-p)E(\mathrm e^{\mathrm itY})$$