average uniform distribution

Question

suppose we have an Uniform distribution $U(0,b)=\frac{u(0)-u(b)}{b}$ where $u(.)$ is the unit step function. The limit $b$, however, is a non-negative random variable with some tail probability.

Is there a way to compute the expectation $\mathbb{E}\{U(0,b)\}$ i.e., to average out the randomness of $b$?

I am struggeling.

Answer 1

You can compute it a Monte-Carlo simulation, if the practical result with approximate precision is all that matters.

Otherwise, the expectation of the uniform variable is $b/2$ hence if you have the probability density function of $b$ it should suffice to compute the expectation of $b/2$.

Answer 2

We have a non-negative random variable $B$ and a random variable $U$ which, conditioned on $B=b$, has uniform distribution on $(0,b)$. It is trivial that $\mathbb E[U\mid B=b] = \frac12 b$ and therefore $\mathbb E[U\mid B] = \frac12B$. However, we cannot compute $\mathbb E[U]$ without knowledge of the distribution of $B$.