I need to compute a conditional probability distribution as described below for my research.

In $(mathbb R^2,||cdot||_2)$, I have a random vector $underline{z}$ with uniformly distributed angle and $Z=||underline{z}||$ following Erlang distribution with $k=2$ and scale parameter $mu$, i.e. with the density function $f_Z(z)=frac{z}{mu^2}e^{-frac{z}{mu}}$. I have another normal random vector $underline{y}$ independent of $underline{z}$. I’m interested in the resultant vector $underline{x}=underline{y}+underline{z}$ and want to compute the conditional distribution of $X=||underline{x}||$ given $aleq||underline{z}||leq b,0leq a<bleqinfty$, to be specific, the complementary cumulative distribution function $overline{F}_{X|Z}(x|(a,b))=P(X>x|aleq Zleq b)$. Solutions for special cases where $Zleq c$ or $Zgeq c$ for any $c>0$ would be sufficient for my research if they are easier to solve.

Following is my attempt. Given a fixed $Z=z$, since $underline{y}$ is normal, $X$ follows the noncentral $chi$ distribution with $k=2$ and non-centrality parameter $lambda=z$, i.e. $f_{X|Z}(x|z)=xe^{-frac{x^2+z^2}{2}}I_0(xz)$, where $I_0(x)=frac{1}{pi}int_0^pi e^{xcosalpha}dalpha$ is a modified Bessel function of the first kind. Then the density function of the conditional distribution is

$$f_{X|Z}(x|(a,b))=frac{int_a^b f_Z(z)f_{X|Z}(x|z)dz}{int_a^b f_Z(z)dz}$$

The denominator $int_a^b f_Z(z)dz=gamma(2,frac{b}{mu})-gamma(2,frac{a}{mu})$ where $gamma$ is the lower incomplete gamma function.

Change the order of integration, the numerator is

$$begin{align}

int_a^b f_Z(z)f_{X|Z}(x|z)dz & = frac{1}{pi}int_a^bfrac{z}{mu^2}e^{-frac{z}{mu}}xe^{-frac{x^2+z^2}{2}}int_0^pi e^{xcosalpha}dalpha \

& = frac{x}{pimu^2}e^{-frac{x^2}{2}}int_0^pi e^{frac{1}{2}(frac{1}{mu}-xcosalpha)^2}int_a^b ze^{-frac{1}{2}(z+frac{1}{mu}-xcosalpha)^2}dzdalpha \

& = frac{x}{pimu^2}e^{-frac{x^2}{2}}int_0^pi e^{frac{beta^2}{2}}left(e^{-bar{a}^2}-e^{-bar{b}^2}+sqrt{frac{pi}{2}}left(operatorname{erf}bar{a}-operatorname{erf}bar{b}right)right)dalpha

end{align}$$

where $beta=frac{1}{mu}-xcosalpha$, $bar{a}=frac{a+beta}{sqrt{2}},bar{b}=frac{b+beta}{sqrt{2}}$, $operatorname{erf}$ is the error function.

Then I got stuck at the second integral. I am looking for an analytical expression of $f_{X|Z}(x|(a,b))$. I tried numerical integration and compared it to a simulation using matlab. The results are as expected.

Finally, what I want is an analytical expression of $overline{F}_{X|Z}(x|(0,c))=P(X>t|Zleq c)=int_t^infty f_{X|Z}(x|(0,c))dx$ and $overline{F}_{X|Z}(x|(c,infty))=P(X>t|Zgeq c)=int_t^infty f_{X|Z}(x|(c,infty))dx$.

Is it possible?