A Discrete Memoryless Channel (DMC) has the following relation between input $X$ and the output $Y$: $$ Y=X+Z, $$ where $X$ lies in the interval $\left(-0.5,0.5\right)$ and $Z$ has uniform distribution on $[-1,1].$ Find the capacity of the channel.
My attempt: $I(X;X+Z)=h\left(X+Z\right)-h\left(Z\right)$. How do I maximise $h(X+Z)$?
Hint: what do you know, a priori (from what you know about $Z$ and the restrictions put on $X$) about the distribution of $Y$? Dou you know that some (families of) distributions maximize the entropy subjected to some restrictions? (for example, the normal distribution maximies the entropy subjected to the restriction of fixed variance). Once you find a candidate distribution that maximizes the entropy for your restricition, check that that bound is attainable by using a proper input distribution.
Update: As noted in the comment, the trivial bound is a uniform on $[-1.5,1.5]$. But that is not attainable. To note this, see that, in the range $-1.5<y<1.5$, $f_Y(y)$ is independent of $f_X(x)$:
$$f_Y(y)=\int_{-0.5}^{0.5} \frac{1}{2}f_X(x)dx=\frac{1}{2} \tag{1}$$
Then, we not only have the restriction that $-1.5\le Y\le 1.5$, but also $(1)$. Then it's straighforward (well...) that the new bound is given by a uniform on the "free" zone. This corresponds to $P(X=0.5)=P(X=-0.5)=1/2$