I have the following formula:
$$\lambda = \lambda(1-F(S-2)) + \frac{r}{c}p$$, where $S, p, r$ and $c$ are constants and $F(.)$ is the CDF function of a random (Poisson distributed) variable, so $F(S-2) =\sum_{x=0}^{S-2} \frac{e^{-\lambda}\lambda^x}{x!}$
I am trying to find $\lambda$ for which the equation holds and I therefore tried the point iteration method.
So I'm deploying the following:
$$\lambda_{n+1} = \frac{\frac{r}{c}p}{F(S-2)}$$, where $$F(S-2) =\sum_{x=0}^{S-2} \frac{e^{-\lambda_n}\lambda_n^x}{x!}$$
But it doesn't seem to converge to a $\lambda$ for which the equation holds...
Anyone an idea why it doesn't converge? Am I doing something wrong? Or could it perhaps be that there is a closed form solution for $\lambda$?
$$F(s-2) =\sum_{x=0}^{S-2} \frac{e^{-\lambda}\lambda^x}{x!}=\frac{\Gamma (s-1,\lambda )}{\Gamma (s-1)}$$ So, what you want is to solve for $\lambda$ the equation $$\lambda\Gamma (s-1,\lambda )= k \qquad \text{with} \qquad k=\Gamma (s-1) \frac r c p$$ or better, to find the possible zeros of function $$f(\lambda)=\lambda\Gamma (s-1,\lambda )- k$$ $$f'(\lambda)=\Gamma (s-1,\lambda )-e^{-\lambda } \lambda ^{s-1}$$ and then Newton iterates are $$\lambda_{n+1}=\frac{\lambda_n \left(\lambda_n^s-e^{\lambda_n } k\right)}{\lambda_n ^s-e^{\lambda_n } \lambda_n \Gamma (s-1,\lambda_n )}$$ The problem is to find a reasonable $\lambda_0$.