Past exam of the mathematics course of the University of Cambridge / 2012 / ia / Paper 2 / 9F / i / Solution

The moment-generating function is $M_X(t)=\mathbb{E}[e^{tX}]$ at real $t$ for which that expectation is finite; $M_X(0)=1$. For independent random variables, the joint law factorizes, soThis is the transform form of independence. Its ordinary finite-valued interpretation applies where both factors are finite, with the transform of a zero multiple equal to $1$.