I have one set of discrete data plotted below. I suspect the data follows Poisson distribution with mean $\mu \simeq 400$ and standard deviation $\sigma \simeq \sqrt{400} = 20$.

$$ n(k) = \frac{\mu ^k}{k!} \exp (- \mu)\ \ \ (\mathrm{eq}.1)$$

But how can I confirm this? I tried numerically to calculate (eq.1) but, as you expect, it overflowed.

enter image description here


Central Limit theorem - a Poisson distribution with a large mean is extremely similar to a discretized normal. With standard deviation $20$, each of those integer points will represent a $z$-score range of $.05$ - from $-0.025$ to $0.025$ at $400$, from $0.025$ to $0.075$ at $401$, and so on. The peak at $400$ should be a probability of about $\frac1{20\sqrt{2\pi}}\approx 0.02$.

You're getting numbers that are nearly twice that. Empirically, it looks like your hypothesized model is wrong, and the data is clustered tighter than a Poisson distribution would be.

  • $\begingroup$ Thank you. It helped me a lot. I have to reconstruct the model though. $\endgroup$ – ynn Jan 13 at 8:39
  • $\begingroup$ On that, there's a quantitative way to check - you should have enough data there to estimate the variance. $\endgroup$ – jmerry Jan 13 at 8:55
  • $\begingroup$ I repeated the calculation 500 times but the result didn't change. After some research, it turned out that the model (actually this is Erdos-Renyi random graph model) was not wrong but my estimation "it should always follow Poisson distribution" were incorrect. Now I have what I wanted. Thank you again. $\endgroup$ – ynn Jan 13 at 10:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.