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