I am trying to solve the following problem and have not been able to. Any help would be much appreciated. The problem is explained below.
$$P(\theta_{B})=\int_0^{\Theta_{B}}\frac{\theta_{\gamma}}{\sigma_{j}^2}e^{-\frac{\theta_{\gamma}^2 + \theta_{\beta}^2}{2\sigma_{j}^2}}I_{0}\left(\frac{\theta_{\gamma}\theta_{\beta}}{\sigma_{j}^2}\right)d\theta_{\gamma}$$
where:
$$(\Theta_{B}, \theta_{\beta}, \sigma_{j}, P(\theta_B) \in \Re)$$ and $I_{0}(z)$ is the modified Bessel function of the first kind and zeroth order. $$I_0(z)= \sum_{k=0}^{\infty} \frac{\left(\frac{1}{4}z^2\right)^k}{k!^2}$$
$\theta_{\beta}$ and $\sigma_{j}$ can be expressed as ratios of $\theta_{y}$: $$x_\beta=\frac{\theta_{\beta}}{\Theta_{B}}, y_j=\frac{\sigma_{j}}{\Theta_{B}}$$ $$0\leq (x_{\beta},y_j)\leq 1$$
I want to be able to calculate $y_j$ when $\Theta_B, x_{\beta},$ and $P(\theta_B)$ are known so that $x_{\beta}$ can be plotted against $y_j$ for a known $P(\theta_B).$