We know the formula, f(A|B) = f(A,B)/f(B). In my mind the left hand side makes sense since we are basically taking a 2D distribution to a 1D distribution(well it does not necessarily integrate to one but you get the point) by just limiting the value that one of the axes can take.
Interlude; if we integrate only f(A|B) this then gives us P(A|B) right?
f(B) is the marginal density of B, which I understand to be: A takes on all the values and we are just summing f(A,B) for all values of A, such that a new pdf is formed which integrates to 1, this also is 1D. So for example if we have variance decreasing as A gets larger in f(A,B) then f(B) will have a lower variance than f(B|A=minimum(A))
And f(A,B) is then just the 2D probability density.
I think I am just not so clear on what exactly happens when you divide a 2D function by a 1D function and therefore I cannot visualise it.