Product of a discrete and absolutely continuous random variable, part deux

I have the independent random variables $U\sim N(0,1)$, $V\sim N(0,1)$, $W\sim b(1,1/2)$

I define $X=WU + (1-W)(V+1)$. I need to determine that $X$ is absolutely continuous, and determine a density function of $X$. I have the same problem as with my earlier question: I really don't know how the joint distribution is defined for a discrete and an absolutely continuous variable.

Bonus question: I also need to determine if $UW$ is a discrete variable. I'm thinking yes, since ${UW=0}\supseteq{W=0}$, and $P({W=0})=1/2$.

• If anyone else could explain the given answer, that would be appreciated! – Elswyyr Aug 5 '14 at 15:17
• Did the link help? – Did Aug 7 '14 at 6:53

$$f_X(x)=\tfrac12f_U(x)+\tfrac12f_{V+1}(x)=\tfrac12f_U(x)+\tfrac12f_{V}(x-1)$$ Note that $UW$ is neither purely discrete nor purely continuous since its distribution has an atom of mass $\frac12$ at $0$ and the rest of the distribution is a densitable measure of mass $\frac12$.