Drawing random realisation from quantity with poisson error How would I draw a random realisation of a variable with an upper and lower error determined from Poisson statistics using the Gehrels 1986 formula?
See: http://adsabs.harvard.edu/abs/1986ApJ...303..336G
For example, I have a count of:
$x = 3^{+5.9}_{-1.3}$
and I want to draw many random realisations of this to use in a Monte Carlo simulation. How do I do this,  please?
If it were a Gaussian variable I'd use a random Gaussian (or split-normal) distribution centred on $\mu=3$ with a standard deviation $\sigma^+=5.9$ and $\sigma^-=1.3$.
Thanks for any help here.
 A: Since you give no information how the confidence interval should affect your simulation (said confidence interval being something you have arbitrarily chosen via choosing its probability cutoffs), I see only two possible answers to your question.
The first is that you should just use the rate which was observed as the parameter for a random variable with a Poisson distribution (since the observed rate is an unbiased estimator for $\lambda$).
The second and more interesting possibility is Bayesian in nature. Given an observation of $k$ events, the probability distribution for $\lambda$ is a gamma distribution with parameters $\alpha = k+1$ and $\beta = 1$. Therefore, you can use a two-step method: generate a value for $\lambda$ from that distribution, and then generate a random value for $k$ from the Poisson distribution with that choice for $\lambda$. (I suppose if you really want your choice of confidence interval to have some effect, you could limit the possible values for $\lambda$ generated by that gamma distribution to your confidence interval, by rejecting any values outside of the interval.)
