# subscript notation in conditional probability

$X$ 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?

• Surely MathStackExchange has a standard wikified answer for the seemingly endless stream of "what does this conditional probability notation mean?" – Eric Towers Jun 17 '14 at 5:11
• @EricTowers sorry, can't find anything directly related. – user13107 Jun 17 '14 at 5:49
• What is your source for the notation $p_X(x|y)$? – Did Jun 17 '14 at 6:35
• @Did see eq. 3.3 here books.google.com/… – user13107 Jun 17 '14 at 6:45
• Thanks. Note that this is hardly a probability theory textbook. – Did Jun 17 '14 at 6:48

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