$X$ 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?

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

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


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