The scaled cumulant-generating function is
$$
\frac1t\log G(\lambda)=\mu(e^\lambda-1).
$$
By <Cramér theorem>, its Legendre transform is stationary at $e^{\lambda_*}=x/\mu$ for $x\geq0$. Thus
$$
\boxed{I(x)=x\log\frac x\mu-x+\mu},
\qquad x\geq0,
$$
with $I(x)=+\infty$ for $x<0$ and the continuous convention $0\log0=0$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 353 / 4 / a / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-353/4/a/ii/solution