I'm trying to calculate the variance of a random variable, which has the following pdf: $\displaystyle f_X(x) = \frac{\exp\left(\kappa\cos(x)\right)}{2\pi I_0(\kappa)}$, where $\kappa$ is a fixed parameter, $I_0(\cdot)$ is the modified Bessel function of the first kind, and $x\in[-\pi, \pi]$.
It is simple to show that $E\{x\} = 0$. So, an equivalent problem is to calculate the following definite integral:
$C = \displaystyle \int_{-\pi}^{\pi}x^2\exp\left(\kappa\cos(x)\right)dx$.
Any clues?
(By the way, we already know that $\displaystyle \int_{-\pi}^{\pi}\exp\left(\kappa\cos(x)\right)dx = 2\pi I_0(\kappa)$, if that might be useful for the solution.)
Thanks!