Find the distribution function of $Z=YX_{1}+(1-Y)X_{2}$ , Y Bernoulli random variable

Question

Let Y a Bernoulli random variable with parameter p and let $X_{1}$ and $X_{2}$ random variables with distribution functions $F_{1}$ and $F_{2}$, respectively. Y, $X_{1}$, $X_{2}$ are independent.

Proof that the distribution function of

$Z=YX_{1}+(1-Y)X_{2}$

is

$F=pF_{1}+(1-p)F_{2}$

(Don't use characteristic functions, generating moment functions).

I really don't have idea.

I know how find the distribution function of W =U+V, but I don't have idea about $W=UV$ or similar

In the notes of my professor's class he has not explained anything similar

Answer 1

$P(YX_1+(1-Y)X_2 \leq x)=P(Y=0) P(X_2 \leq x)+P(Y=1)P(X_1 \leq x)=(1-p)F_2(x)+pF_1(x)$. In the first equality I have written the event $(YX_1+(1-Y)X_2 \leq x)$ as the disjoint union of $(Y=0,YX_1+(1-Y)X_2 \leq x)$ and $(Y=1,YX_1+(1-Y)X_2 \leq x)$, then used independence.