Marginal Distribution

Marginal Distribution:

It’s a function that gives the probability based on only subset of the variables.¹. For example,

Or in mathematical way, for discrete

$$Pr(X = x) = \sum_y Pr(X = x, Y = y) = \sum_y Pr(X = x | Y = y) Pr(Y = y)$$

and for continuous

$$p_X(x) = \int_y p_{X,Y}(x,y) dy = \int_y p_{X|Y}(x|y)p_Y(y) dy$$

and it can also be written as Expected Vaue

$$p_X(x) = \int_y p_{X|Y} (x|y) p_Y(y) dy = \mathbb{E}_Y[p_{X|Y} (x|y)]$$