Find the density function of the product of two uniformly distributed

Question

$X\sim R(0,1)$ and $Y\sim R(0,1)$ (X and Y are uniformly distributed on the interval $[0,1])$

I need to find the density function of W when $W=X\cdot Y$

Can anyone help me?

Answer 1

Hint (preassuming that $X$ and $Y$ are independent):

For $w\in(0,1)$ we have: $$F_{W}(w)=\int\int1_{(-\infty,w]}(xy)f_X(x)f_Y(y)dxdy=\int^w_0\int^{w/y}_0dxdy$$

Work this out and find PDF $f_W(w)$ as derivative of $F_W(w)$.