Question about how to obtain the value of f X|Y(x|y)

Question

In the following example question (from Bertsekas, edition 1), i have one question:

Why the value of fY|X(y|x) is 1/2?

Is it because Y is Y|X is either 0 or 1/6 (50% probability), or because some other reason? The confusion point for me is that Y = X + W. Then, should fY|X（y|x) should be similar to fx(x)?

I have always have difficulty when I come to look at conditional PDFs for they are not as easily (at least from my part) to understand at this point.

Answer 1

Given $X=x$, $Y$ is uniformly distributed in interval $[x-1,x+1]$ of length $2$, so

\begin{equation} f_{Y|X}(y|x)= \begin{cases} 1/2, & x-1\leq y\leq x+1\\ 0, &otherwise \end{cases} \end{equation}