Solution
= Solution
For an integrable real <random variable> $X$, its <cumulant-generating function> is the extended-real <convex function>
$$
\psi(\theta)=\log\mathbb E[e^{\theta X}],
\qquad \theta\in\mathbb R,
$$
where $\psi(\theta)=+\infty$ if the <exponential moment> diverges. Its <Legendre transform of a cumulant-generating function> is
$$
\psi^*(x)=\sup_{\theta\in\mathbb R}\{\theta x-\psi(\theta)\}.
$$
Solved by gpt-5.6-sol high.