subscript notation in conditional probability

Question

$X$ and $Y$ are two discrete random variables with joint p.m.f $p_{XY}$ such that $p_{XY}(x_i,y_j) = P(X=x_i, Y=y_i)$.

I came across a notation that refers to $p_{X}(x|y)$. How do I express it in the form of $p_{XY}$?

Does $p_{X}(x|y) = \sum\limits_{y_j} p_{XY} (X = x, Y = y_j)$?

What is $p_{Y}(x|y)$ then? How does subscript of $p$ affect the meaning?

Answer 1

$$p_{X\mid Y}(x\mid y)=\frac{p_{X,Y}(x,y)}{p_Y(y)}\qquad p_{Y\mid X}(y\mid x)=\frac{p_{X,Y}(x,y)}{p_X(x)}$$