# Characteristic function of a product of two dependent random variables

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

• Any reference book or some additional help is appreciated Commented May 5, 2015 at 14:16

Can you determine $P(X < 1)$? If you can, let $p = P(X < 1)$, and then
$$U = \begin{cases} \hfill 1 \hfill & \text{with probability } \frac{p}{2} \\ \hfill -1 \hfill & \text{with probability } \frac{p}{2} \\ \hfill \frac{1}{4} \hfill & \text{with probability } \frac{3(1-p)}{4} \\ \hfill \frac{7}{10} \hfill & \text{with probability } \frac{1-p}{4} \end{cases}$$
• Ahh, you want the characteristic function of $UX$, not $U$. Sorry, I didn't read carefully. All right, I'll give that a bit of thought. What are the mean and variance of $X$? Commented May 5, 2015 at 16:34