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

Elswyyr
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1 Answers1

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$$f_X(x)=\tfrac12f_U(x)+\tfrac12f_{V+1}(x)=\tfrac12f_U(x)+\tfrac12f_{V}(x-1)$$ 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 $\frac12$.

Did
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  • 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. – Elswyyr Aug 05 '14 at 14:22
  • You cite "absolutely continous", "density function", "joint distribution", but you are allowed no measure theory notions? What is this course? – Did Aug 05 '14 at 14:24
  • This is the freshman "Introduction to Probability Theory" course at Aarhus University. Maybe things are structured or named differently in US universities? – Elswyyr Aug 05 '14 at 14:26
  • How do they suggest to compute PDFs in general? – Did Aug 05 '14 at 14:28
  • 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. – Elswyyr Aug 05 '14 at 14:30
  • No general definition? – Did Aug 05 '14 at 14:48
  • Let us continue this discussion in chat. – Elswyyr Aug 05 '14 at 14:49
  • Most general definitions I've got is that the function must be non-negative and integrate to 1 over the reals. Oh, and that the probability of an event in a subset is the integral of the PDF over the subset. And that is about as general as it goes. – Elswyyr Aug 05 '14 at 15:17
  • These are not a definition, at most properties. How do you determine/check that some function $h$ is the PDF of the random variable $X$? – Did Aug 05 '14 at 15:30
  • Well, if I know the distribution function of the variable, I can find the derivative. Otherwise, well, I don't. – Elswyyr Aug 05 '14 at 15:32
  • To sum up, you have no definition of the PDF of a random variable at hand? This might be problematic if one wants to solve exercises involving PDFs... :-) – Did Aug 05 '14 at 15:34
  • Indeed. Unfortunately, this is an exam set from a previous year, so it is stuff I need to be able to answer. – Elswyyr Aug 05 '14 at 15:36
  • Then you might find this useful. – Did Aug 05 '14 at 15:40