How can I find P[X ≥ 2Y] from the joint pdf?

Question

Let X and Y have the joint pdf $f(x,y) =xye^{−x−y}$ $(x > 0,y > 0)$

Find $P[X ≥ 2Y]$.

I really don't know what can I do for this...

Answer 1

Hint:

$$P(X\geq 2Y)=\mathbb E\mathbf1_{X\geq2Y}=\int\int\mathbf[x\geq2y]f(x,y)dxdy$$where $[x\geq2y]$ is a function $\mathbb R^2\to\mathbb R$ that takes value $1$ if $x\geq 2y$ and value $0$ otherwise.

Answer 2

I would start with

$$\int_0^\infty\int_{2y}^\infty f(x,y)dxdy.$$

That is

$$\int_0^\infty ye^{-y}\bigg(\int_{2y}^\infty xe^{-x}dx\bigg)dy$$

Compute the integral in () then see where it goes?

Answer 3

Hint:

$$\int_{0}^{\infty} \int_{2y}^{\infty} xye^{-x-y}dxdy = \frac{7}{27}$$