# Simple approximation of tail surprisal of poisson distribution

I want to determine (in an algorithm) the approximate surprisal of getting an outcome "as extreme as $k$" from the $Poisson(\lambda)$ distribution.

My original plan was to use $-log_2(1/2-|1/2-F_{Poisson(\lambda)}(k)|)$ but calculating $F_{Poisson(\lambda)}(k)$ requires the incomplete Gamma function which uses an iterative numerical algorithm. I am hoping to find a simple approximation (of the CDF, or even better, of the surprisal itself) that will allow me to speed things up, because I'm doing this for each pixel in a video.

The fact that erf has such simple rational function approximations made me think $F_{Poisson}$ might have something similar (though we have a 2D domain here so perhaps that was wishful thinking.) As a last resort, I might create a 2D lookup table with a log-log-spaced grid and use bilinear interpolation (on the surprisal directly).

## Error

The surprisal only needs to be accurate to within about 1 bit for all surprisals less than about 15 bits, translating to the requirement that $F_{Poisson(\lambda)}(k)$ is approximated to within about 50% of the distance from whichever of 0 or 1 is closer until the remaining tail probability is about $2^{-15}$. I'm just making up these specifications to give a rough idea.

## Previous related questions

My original question on SO involved a very skew Binomial distribution (of unknown N), but it is so skew that I lose nothing by approximating that with the Poisson distribution, thereby getting rid of one domain variable.

To be honest, I don't feel comfortable asking such an asker-specific question here, but the discussion here is what encouraged me to post it anyway. Any pointers or advice would be helpful.

For closure:

$$F_{Poisson(\lambda)}(x) \approx \Phi(z)$$

where

$$z = \frac{x-\lambda+\frac{2}{3}}{\sqrt{\lambda(1+0.68w+(0.12-signum(w)0.0325)w^2)}}$$

where

$$w = \sqrt{\frac{x+\frac{1}{2}}{\lambda}}-1$$

Using this approximation, the surprisal can then be calculated directly from $z$.