Solution
= Solution
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.