Fun statistical observation: If X follows a Chi square distribution with k degrees of freedom, while the MGF E[exp(tX)] of X doesn't exist for t>1/2, I computed that, for k=3, and the cardinal hyperbolic sine sinhc(u):=sinh(u)/u that basically behaves like exp(u)/2u if u is large, then E[sinhc(tX)] = exp(t^2/2) for all values of t. This can be found from the Taylor development of sinhc and the moments of X (see e.g. https://statproofbook.github.io/P/chi2-mom.html). I'm wondering if this is a well known fact? #statistics #chisquare #mgf #nospecificreason