My question is basically what the title says. I am solving some exercises in statitics and was told no knowledge of measure theory is required but I am not sure if that is the case.
I've been looking into Lebesgue measures but I cannot find a definition of what it means that a random variable has Lebesgue density given by some formula.
For example a question where I've been stuck because of this:
Let $Y$ be a random variable with a Lebesgue density, and let $Z$ be a discrete random variable with (probable) probabilities $\{ z_k \mid k \}$. Furthermore, let $Y$ and $Z$ be independent. Justify that $X = Y + Z$ also has a Lebesgue density and calculate it.
Thank you!
Lebesgue density will refer to a "density" with respect to the Lebesgue measure. If you don't know much measure theory, then just read this as: "$Y$ has a probability density" (a function $f$ such that $\mathbb{P}(Y\in A)=\int_A f(x)\,\mathrm{d}x$ for all events $A$).
You can view "$\mathrm{d}x$" as $\mathrm{d}\mu$ for the Lebesgue measure $\mu$. All $\mu$ is is the natural way to associate a size to subsets of the real line, most importantly with $\mu([a,b])=b-a$. It is the length measure, if you like.
For further reading look at Radon-Nikodym derivatives. In many cases we cannot explicitly find this density function, however to show it exists we just need to show the following:
A precise definition of measure zero is a little technical (if you haven’t done any measure theory). It’s worth having some understanding of the basics in my opinion.