Expectation of Truncated Rice distribution I'm searching for the first order moment of a truncated Rice distribution.
More precisely, i'm searching for the result of
$$\frac{1}{F(l|\nu,\sigma)}\int_{0}^{l}xf(x|\nu, \sigma)dx,$$
where
$$f(x|\nu, \sigma)=\frac{x}{\sigma^2}exp\left(\frac{-(x^2+\nu^2)}{2\sigma^2}\right)I_0\left(\frac{x\nu}{\sigma^2}\right),$$
and
$$F(x;\nu,\sigma)=1-Q_1(\nu/\sigma, x/\sigma),$$
with $Q_1$ the marcum Q-function and $I_0$ the modified Bessel function of the first kind and order 0.
Integrating by parts the first formula, it seems to me that this resumes to integrating 
$$\int_0^lQ_1(b,ax)dx.$$
I've came accros this document and the formula (60) is close from what i need, but for $l=\infty$.
I'm stuck there. Any clue ?
Thanks for your help : )
 A: One imperfect answer here, based on an approximation of the Marcum Q-function.
Starting with an integration by part, we get
$$\int_{0}^{l}xf(x|\nu, \sigma)dx=\left[x(1-Q_1(\nu/\sigma, x/\sigma))\right]_0^l-\int_0^l(1-Q_1(\nu/\sigma,x/\sigma))dx\\
=lQ_1(\nu/\sigma,x/\sigma)+\int_0^lQ_1(\nu/\sigma,x/\sigma).$$
All the hard work is now to integrate $\int_0^lQ_1(\nu/\sigma,x/\sigma)$.
I did not find any analytical solution to this. But in this paper, the Marcum Q-function is approximated using :
$$Q_1(a,b)\approx\tilde{Q}_1(a,b)=\exp(-e^{p}b^{q})$$
where $p$ and $q$ are polynomials depending on $a$ (how to find the polynomials is described in the reference). It turns out that this function now integrates well : 
$$\int_0^l\tilde{Q}_1(a,x)dx=\frac{(e^p)^{-1/q}(\Gamma(\tfrac{1}{q})-\Gamma(\tfrac{1}{q},l^qe^p))}{q},$$
with $\Gamma(a)$ the gamma function, and $\Gamma(a,x)$, the incomplete gamma function.
Taking care of the change of variable (dividing with $\sigma$), we now have an approximate solution to $\int_{0}^{l}xf(x|\nu, \sigma)dx$.
