I have a Poisson Process $N_t$ of rate 2/second. I'm asked to approximate the probability that more than 7250 events happen in the first hour. This probability is
$$ P(N(0, 60^2] > 7250) $$ and as $N_t$ is a Poisson Process of rate 2, we have $$ N(0, 60^2] \thicksim Poisson(2*60^2) \\ \implies \\ P(N(0, 60^2] > 7250) \\ = 1 - P(N(0, 60^2] \leq 7250) = 1 - e^{-2*60^2} * \sum_{i=0}^{7250} \frac{(2*60^2)^i}{i!} $$
Now, this involves numbers too large to evaluate. Normally in this case, I would calculate probabilities in log space but this seems impractical here due to the sum terms. I'm asked only an approximation so I could use an approximation for $log(n!)$ though.
I do know $\mathbb{E}[N(0, 60^2]] = 7200$ and $var(N(0, 60^2]) = 7200$ by properties of Poisson distributions. But does that help me in any way to get a decent approximation?
I ran into similar problems using the sum of inter-arrival times, which is Gamma distributed.
How would I get to a decent approximation here?