Given $X\sim U(0,2)$ and $Y\sim U(0,3)$ which are length and width of a rectangle (respectively). I want to find the probability of the area of the rectangle less than 1.
The hint is the joint density $f(x,y) = 1/6$ for $0\leq x \leq 2$ and $0 \leq y \leq 3$.
So, f(x,y) = 0, otherwise.
The answer is 0.4653, but never explained why.
Best Answer
The joint distribution is depicted below together with the function $y=\frac1x$. $(X,Y)$ will have to fall in the pink area so that $XY<1.$
The probability we are looking for is the pink area times $\frac16$.
$$P(XY<1)=\frac16+\frac16\int_{\frac13}^2\frac1x\ dx=\frac16+\frac{\ln(6)}6\approx.4653.$$