Distribution of a function of a uniform random variable.

Question

I ran across this example the other day and was surprised at how stumped I was. Suppose $U$ is a uniform random variable on the interval $[0,1]$. Let $F = \frac{1}{U+3}$. What is:

$\operatorname{Cov}(U,F)$?

$f_{F\mid U}(x\mid u)$?

$E[F\mid U=u]$?

Answer 1

Since $F$ is an invertible function of $U$, the conditional distribution of $F$ given $U=u$ has only a single point and is hence degenerate.

For the first one, note that $\text{Cov}(U,F)=E(UF)-E(U)E(F)$.