What's the pdf of the product of one continuous random variable and a discrete one?

Question

Let $X_1$ be a random variable with bimodal distribution i.e. $ p(X_1 = x_1) = 1/2 ~\delta(x_1 - \beta ) + 1/2 ~\delta(x_1 + \beta ) $ with $\beta > 0$. And let $X_2$ be a uniform distributed random variable in the range $[a, b]$ also with $a, b > 0$. What is the pdf of $Y = X_1 X_2$? Is there a book treating this mixed case? I have only found references (the algebra of random variables by Springer) for the case in which both $X_1$ and $X_2$ are discrete or continuous.

Answer 1

Assuming $X_1$ and $X_2$ are independent, conditioned on $\{X_1=\beta\}$ it is clear that $X_1X_2$ is uniformly distributed on $(\beta a,\beta a)$ and similarly conditioned on $\{X_1=-\beta\}$, $X_1X_2$ is uniformly distributed on $(-\beta b,-\beta a)$. Since $\mathbb P(X_1=\beta)=\frac12=\mathbb P(X_1=-\beta)$, it follows that the density of $X_1X_2$ is given by $$ f_{X_1X_2}(t) = \frac1{2\beta(b-a)}\left(\mathsf 1_{(-\beta b,-\beta a)}(t)+\mathsf 1_{(\beta a,\beta b)}(t) \right). $$