Let $X$ be a random variable from a uniform distribution $\mathrm{Unif}[1,2]$ and $U = h(X) = \begin{cases}1, & X < 5/4 \\ 0, & \mathrm{else}\end{cases}$
I want to find the covariance by using the expectation values of $X$, $U$ and $UX$.
I am having trouble finding the expectation value $\mathbb{E}[UX]$ since I'd have to calculate
$$\int_{-\infty}^\infty \int_{-\infty}^{\infty} ux f_{UX}(u,x) \mathrm{d}u\mathrm{d}x$$
but I can't get the joint pdf. Since $u$ depends on $x$ in a deterministic way, does it even make sense to talk about a joint pdf?
On the other hand, the expectation values of $X$ and $U$ are pretty straight forward being $3/2$ (mean of uniform distribution) and $1/4$ (the probability that $X \leq 5/4)$ respectively.
Thank you
The random variable $UX$ does not admit a density as it has a positive probability mass in
$$\mathbb{P}[UX=0]=0.75$$
thus
$$\mathbb{E}[UX]=1.5\times0.25=0.375$$
concluding:
$$\mathbb{Cov}[U,X]=0.375-0.25\times1.5=0$$