So in this problem I have two i.i.d random variables $X$ and $Y$, which are uniformly distributed over interval $[0,1]$.
I know that $S = X + Y$. To find the density of $S$, I find the convolution of $X$ and $Y$. However I am struggling conceptually to understand what $p_{X \vert S}(x \vert s)$ is, for a given real value of $S$.
The conflict is that $X$ is defined as a uniform over [0,1], so it seems that the pdf should remain just that. However clearly if we condition on $S$ being 0.5 for example, then $X$ cannot sample values greater than that.
I am inclined to say
$p_{X \vert S}(x \vert s) = \begin{cases} \frac{1}{s} & \mathrm{for}\ 0 \le x \le s, \\[8pt] 0 & \mathrm{otherwise}\ \end{cases} $
But this is clearly wrong if s > 2, or s < 0, and I am not sure it is correct. Could someone offer advice on how $p_{x \vert S}(x \vert s)$ would be derived?