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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

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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.

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  • $\begingroup$ 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

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