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.
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.)