Let X,Y∼Uniform(0,1)X, Y \sim \text{Uniform}(0, 1)X,Y∼Uniform(0,1) be independent. What is the probability P(XY≤z)P(XY \le z)P(XY≤z) for any 0<z<10 < z < 10<z<1?
z(1−lnz)z(1 - \ln z)z(1−lnz)
zlnzz\ln zzlnz
1−z+zlnz1 - z + z\ln z1−z+zlnz
z(1+lnz)z(1 + \ln z)z(1+lnz)