 Solution

Because $h^{\prime }(x)=-g(x)/x$, the joint density from part (b) can be writtenThe pieces are therefore independent if and only ifThis condition is also sufficient because $h$ is the common marginal density.

Set $c=h(0)$. Taking $z=0$ gives the differential equationSince $h$ is a probability density, $c$ must be positive, and normalization givesConversely this function satisfies the displayed factorization. Thus the exponential characterization by a uniform random split shows thatEquivalently, $g(x)=-x{h}^{\prime }(x)=c^{2}x{e}^{-cx}$, so $X$ has the gamma distribution with shape two and rate $c$.