What is the probability that $$ax^2 + bx + c = 0$$ has real solutions if $a, b, c$ are Poisson variables with common mean $\lambda$?
This seems to require evaluating $P(b^2 - 4ac \geq 0)$, which is the sum $$e^{-3\lambda} \sum_{\substack{a, b, c \geq 0 \\ b^2 \geq 4ac}} \frac{\lambda^{a + b + c}}{a!b!c!}.$$ I'm not even sure how to begin evaluating such a thing though.
The question is straightforward if only one of the three coefficients is random, but even two makes it difficult. I would happily accept an answer to the question when, say, only $b$ and $c$ are random.
