Suppose there are two independent random variables: $\mathbb{A}$ ~$\mathit{N}(0,1)$ [ie standard normal] and $\mathbb{B}$ that is ~Bernoulli with $P(\mathbb{B} = 1) = P(\mathbb{B} = -1) = 0.5$. Find the pdf distribution for $\mathbb{C} = \mathbb{A}\mathbb{B}$.
Independence: $$f_{\mathbb{XY}} = f_{\mathbb{X}}(x)f_{\mathbb{Y}}(y)$$ However, $\mathbb{B}$ has a piecewise pmf as opposed to the continuous pdf of $\mathbb{A}$. Is the above formula still valid for a 'mix' of discrete and continuous variables? Or should I focus on the definition of a CDF: $$F_\mathbb{C}(c) = P(\mathbb{C} \leq c) = P(\mathbb{AB} \leq c)$$ find the CDF and take the derivative?
Thanks!
If $J$ is an interval not containing $0$, $$\mathbb P(AB \in J) = \mathbb P(B=1) \mathbb P(A \in J) + \mathbb P(B=-1) \mathbb P(-A \in J)$$ But since the distribution of $A$ is symmetric, $\mathbb P(-A \in J) = \mathbb P(A \in J)$. We conclude that $\mathbb P(AB \in J) = \mathbb P(A \in J)$, so $AB$ has the same distribution as $A$.