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 $P({W=0})=1/2$.

  • $\begingroup$ If anyone else could explain the given answer, that would be appreciated! $\endgroup$ – Elswyyr Aug 5 '14 at 15:17
  • $\begingroup$ Did the link help? $\endgroup$ – 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 $\frac12$.

  • $\begingroup$ I'm afraid I have no idea why this result is correct, or how you derived it. Furthermore, I haven't taken any measure theory courses yet, so I'm pretty sure a simpler answer is expected of me, of the bonus question at least. $\endgroup$ – Elswyyr Aug 5 '14 at 14:22
  • $\begingroup$ You cite "absolutely continous", "density function", "joint distribution", but you are allowed no measure theory notions? What is this course? $\endgroup$ – Did Aug 5 '14 at 14:24
  • $\begingroup$ This is the freshman "Introduction to Probability Theory" course at Aarhus University. Maybe things are structured or named differently in US universities? $\endgroup$ – Elswyyr Aug 5 '14 at 14:26
  • $\begingroup$ How do they suggest to compute PDFs in general? $\endgroup$ – Did Aug 5 '14 at 14:28
  • $\begingroup$ We have a table of probability density functions of common distributions. We mostly work with independent variables, so finding a joint density usually isn't that bad. This is why I am somewhat lost when it comes to the more specific properties of PDFs. $\endgroup$ – Elswyyr Aug 5 '14 at 14:30

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