Assuming a Poisson distribution, the probability ($\alpha$) that the result will fall within the range $0\ldots k$ is given by the following expression:

\begin{equation} \alpha = e^{-\lambda}\sum_{i=0}^{k}\frac{\lambda^i}{i!} \end{equation}

where $\lambda$ is the expected value (and variance).

In this particular case, the probability ($\alpha$) as well as the range ($k$) is known but the expected value ($\lambda$) is unknown. Is there a analytical solution or close approximation to the value of $\lambda$?

I have so far been solving the problem numerically by minimizing the equation but this is consuming a rather large part of the processing time of my computer which is why I am looking for ways to optimize my code.