Find the probability that the rectangle whose adjacent sides have lengths X and Y has area less than 8.

Question

Let $X$ and $Y$ be the lengths in inches of adjacent sides of a rectangle. Assume that $X$ is a uniform random variable on the interval $(0,8)$, $Y$ is a uniform random variable on the interval $(0,4)$, and that $X$ and $Y$ are independent. Thus, the joint PDF of the random pair $(X,Y)$ is $f(x,y)=f_X(x)f_Y(y)=(1/8)(1/4)=(1/32)$ when $(x,y)\in (0,8)\times(0.4)$, and 0 otherwise. Find the probability that the rectangle whose adjacent sides have lengths $X$ and $Y$ has area less than 8.

Answer 1

Hint: It wants you to calculate $P(XY<8)$, that is: $$P\left(Y<\frac8X\right) =\int_0^8 \int_0^{8/x} f(x,y) dydx $$