Being Y a standard normal random variable and X an independent Bernoulli random variable, what is the characteristic function of YX?

Question

$$Y \sim N(0,1) $$

$$ X \sim b\left(\frac 1 2\right)$$

Knowing that $Y$ and $X$ are independent, how do I compute the characteristic function of $YX$? What do I have to expect? A function that depends on a scalar $t$ or a vector $(t_1,t_2)$ ?

Thanks in advance.

Answer 1

\begin{align} E[\exp(itXY)] &= E[\exp(itXY) \mid X = 0] P(X = 0) \\ &+ E[\exp(itXY) \mid X = 1] P(X = 1) \\ &= E[1 \mid X=0] \frac12 + E[\exp(itY)] \frac12 \tag{$X \sim Ber(1/2)$} \\ &= \frac12\left( 1+\exp\left(-\frac12 t^2\right) \right) \tag{$Y \sim N(0,1)$} \end{align}