Two continuous random variables are said to be jointly continuous if they have a joint probability density function (or equivalently you can state it in terms of absolute continuity with respect to two-dimensional Lebesgue measure). Let $X$ be a random variable uniformly distributed over $[0,1]$, and let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a Borel measurable function. My question is, for what functions $f$ are $X$ and $f(X)$ jointly continuous?
I know $f$ can't be the identity function or the absolute value function, so what can it be? If it's not possible to state a general condition for it, does anyone know of particular examples? I'm just trying to better understand the notion of joint continuity.
Best Answer
Never. The joint distribution of $(X,f(X))$ is always singular to Lebesgue measure.
The random vector $(X, f(X))$ always takes values in the graph of $f$, i.e. the set $G_f = \{(x,f(x)) : x \in \mathbb{R}\} \subset \mathbb{R}^2$. It is an exercise to verify that whenever $f$ is a Borel function, the set $G_f$ is Borel. Then it follows from Fubini's theorem that $G_f$ has measure zero (every vertical section consists of one point and so has measure zero).