Let G be a geometric random variable with success probability p. Now, conditional on G = g, let X = U1U2 . . .Ug be a product of g independent and identically distributed uniform (on [0, 1]) random...

Let G be a geometric random variable with success probability p. Now, conditional on G = g, let X = U1U2 . . .Ug be a product of g independent and identically distributed uniform (on [0, 1]) random variables. For example, if g = 3 then X = U1U2U3 where the U1,U2,U3 are iid U[0, 1] random variables.
Find the unconditional probability density function for X.