Past exam of the probability course of the University of Oxford / 2014 / 2 / a / Solution

Integrating $e^{tx}$ against the Gamma density gives the moment-generating function of a gamma distribution:The change of variables $y=(\lambda -t)x$ reduces the integral to $\Gamma (r)=(r-1)!$ using the gamma function. For $t\ge \lambda$ the defining integral is infinite. In particular the moment-generating function is finite on a neighbourhood of zero.