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For an exercise, I have to find the right inverse of the Poisson CDF $$F_X = e^{-\lambda} \sum_{i=1}^{\lfloor k\rfloor }\frac{\lambda^i}{i!}$$ where the right inverse is: $$F_X^{-1}(p):=\text{inf}\{x \in \mathbb{R} : F_X(x) \geq p\}.$$

Does a closed-form solution exist? How do I start to try to find it?

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