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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 201 / 2 / a / Solution

{title2=$\psi_X(\theta)$}
{wiki=Cumulant-generating_function}

The cumulant-generating function is $\psi_X(\theta)=\log\mathbb E e^{\theta X}$, interpreted as an extended-real <convex function> where the exponential moment may be infinite.

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