Characteristic function of a product of two dependent random variables

Question

If you're given the characteristic function of a continuous random variable, say X, and the distribution of another discreet random variable, say U, which is dependent of X, how do you explicitly find the characteristic function of UX? Consider the case that X is normal random variable while U is a random variable defined as

$$ U=\begin{cases} v & \text{if }X<1 \\ r & \text{ }Otherwise \end{cases} $$

where $$ v=\begin{cases} 1 & \text{with probability }\frac{1}{2} \\ -1 & \text{with probability }\frac{1}{2}% \end{cases} $$

and $$ r=\begin{cases} 0.25 & \text{with probability }0.75 \\ 0.7& \text{with probability }0.25% \end{cases} $$

Answer 1

Can you determine $P(X < 1)$? If you can, let $p = P(X < 1)$, and then

$$ U = \begin{cases} \hfill 1 \hfill & \text{with probability } \frac{p}{2} \\ \hfill -1 \hfill & \text{with probability } \frac{p}{2} \\ \hfill \frac{1}{4} \hfill & \text{with probability } \frac{3(1-p)}{4} \\ \hfill \frac{7}{10} \hfill & \text{with probability } \frac{1-p}{4} \end{cases} $$

and you can continue as usual.