Let $X, Y ∼ U[0, 1]$ and suppose that $X $ and $Y$ are independent.

Question

Let $X, Y ∼ U[0, 1]$ and suppose that $X $ and $Y$ are independent.

Find $P(X ≤ Y ≤ 2X)$

Any ideas on how to get this one started?

Answer 1

$$P(X < Y < 2X) = \int_0^1\int_{y/2}^y \, dx \,dy = \frac{1}{4}$$

Note that the joint density is $0$ outside the unit square and

$$P(X < Y < 2X) \neq \int_0^1\int_{x}^{2x} \, dy \,dx $$

Answer 2

Draw the part of the unit square $[0,1]^2$ that corresponds to this event. It's straightforward to find its area geometrically.