I have been working on this for several days and have been unable to come up with an answer. The problem is very simple to state, but it seems difficult to solve.
A computer draws a number $x$ at random from a uniform distribution between $a$ and $b$. The computer also draws a number $y$ from a normal distribution with mean $m$ and standard deviation $s$. The computer then calculates $z = x + y$. The computer reports $z$ but it does not report $x$ or $y$. Calculate the expected value of $x$ given $z$.
So far, I have noted that the probability distribution function producing $z$ is the convolution of the uniform and normal probability distribution functions producing $x$ and $y$, respectively. And I believe that I need to use Bayes' Theorem to determine the value of $x$ given $z$, but I am stuck soon beyond that point. Any help much appreciated!