2D-Random Variables

If (X,Y) is a discrete random variable with $Pr[Y=y_j]>0$ then,

p_{X/Y}(x/y) = P\left[ X = x_i / Y = y_j \right] = \frac{Pr[X=x_i,Y=y_j]}{Pr[Y=y_j]}=\frac{p_{ij}}{\sum_i p_{ij}}

is the conditional pmf of X given Y. Similarly, the set of values $\frac{p_{ij}}{\sum_j p_{ij}}$ would be the conditional pmf of Y given X.

If (X,Y) is a discrete random variable with $Pr[Y=y_j]>0$ then the conditional d.f of X given Y=y,

F_{X/Y} (x/y) = Pr \left[ X \leq x / Y=y \right] = \frac{1}{\sum_j p_{ij}} \sum_{\{i| x_i \leq x\}} p_{ij}

Similarly, provided that $Pr[X=x_i]>0$ the conditional distribution function of Y given X=x is:

F_{Y/X} (y/x) = Pr \left[ Y \leq y / Y=x \right] = \frac{1}{\sum_i p_{ij}} \sum_{\{i| y_j \leq y\}} p_{ij}

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