> we thinking of cutting slice through a multivariate distribution, and then normalizing. If you have some distribution P(X, Y) where X and Y are both real, then its density is like some landscape (i.e. each x,y is associated with some height). P(X | Y=y) is like a slice through that landscape at Y=y, which leaves a 1-D plot of the density along X. If we re-scale it so that the area under it is 1, then it's a well-formed PDF.

How intuitive!

/s