I have a poisson distribution with lambda = 10, and am asked to find the probability that k $\geq$ 7.
I know that it should equal: $$\sum_{k=7}^\infty e^{-10}\frac{10^k}{k!}$$
From here I can pull out the $e^{-10}$ and, I know that the rest is the taylor series for $e^{10}$ minus the first 6 values. However, to get the final answer that means calculating the first 6 values. This method isn't feasible for large initial values of k. Is there a quick way to compute this probability? Are there any mistakes in what I have so far?
The best closed form form for the partial sums of the first $n$ terms known will be even harder to compute, namely we have the formula
$$\sum_{k=0}^n\frac{x^k}{k!}=\frac{\Gamma(n+1,x)}{\Gamma(n+1)}e^x$$
Where $\Gamma(s,x)$ is the upper incomplete gamma function defined by
$$\Gamma(s,x)=\int_x^\infty t^{s-1}e^{-t}dt$$
And of course $\Gamma(s)=\Gamma(s,0)$. So, no, for large initial values of $k$ your best bet is going to be simply computing the partial sum the terms from 1 to your starting point.