= Reciprocal moment of a noncentral chi-squared variable
{title2=$\mathbb E\frac1Q=\int_0^\infty(1+2t)^{-m/2}e^{-\lambda t/(1+2t)}dt$}
For $Q\sim\chi_m^2(\lambda)$ with $m>2$, completing the square in its Gaussian representation gives $\mathbb Ee^{-tQ}=(1+2t)^{-m/2}e^{-\lambda t/(1+2t)}$. Integrate the identity $1/q=\int_0^\infty e^{-tq}dt$ and use <Tonelli theorem> to obtain the displayed reciprocal moment. It is strictly decreasing in $\lambda\geq0$, since its positive integrand is strictly decreasing for every $t>0$. At $\lambda=0$ the integral equals $1/(m-2)$, proving finiteness and the sharp maximum over noncentrality parameters.
Back to article page
