Probability and Joint Densities

Densities spread probability over the plane

A worked density

Marginals and expectations

Integrals play the sum's role

Before using $f(x, y) = c\, x y^2$ as a joint density on the unit square, what must hold?

For $f = 6xy^2$ on the unit square, what is $P(X < Y)$?

From the joint $f = 6 x y^2$ on the unit square, how is the marginal density of $X$ found?

With marginal $f_X(x) = 2x$ on $[0, 1]$, give $E[X]$, as a fraction like 2/3.

For $f = 6xy^2$ on the unit square, give $P(0 \le X \le 1/2)$, as a fraction like 1/3.