fitting Poisson distribution to data in python

Question

I have data distribution that I want to fit Poisson distribution to it. my data looks like that:

I try to fit :

 mu = herd_size["COW_NUM"].mean() 
ax=sns.displot(data=herd_size["COW_NUM"], kde=True)
ax.set(xlabel='Size',title='Herd size distribution & poisson distribution')
plt.plot(np.arange(0, 2000, 80), [st.poisson.pmf(np.arange(i, i+80), mu).sum()*len(herd_size["COW_NUM"])
                                  for i in np.arange(0, 2000, 80)], color='red')
#every bin contain approximatly 80 observes
plt.show()

but I get something not at the same scale:

UPDATE I try to apply negative binom distribution with the code:

n=len(herd_size["COW_NUM"])
p =herd_size["COW_NUM"].mean()/(herd_size["COW_NUM"].mean()+2) 
ax=sns.displot(data=herd_size["COW_NUM"], kde=True)
ax.set(xlabel='Size',title='Herd size distribution & geometry distribution')
plt.plot(np.arange(0, 2000, 80), [st.nbinom.pmf(np.arange(i, i+80), n,p).sum()*len(herd_size["COW_NUM"])
                                  for i in np.arange(0, 2000, 80)], color='red')
#every bin contain approximatly 80 observes
plt.show()

but I got this: nbinom

Answer 1

Your plot is (at least approximately) correct, the problem is with modeling your data as Poisson. As lambda grows large the Poisson looks more and more like a normal distribution — see this plot from Wikipedia. A Poisson distribution has its variance equal to its mean, so with a mean of around ~240 you have a standard deviation of ~15.5. The net result is that outcomes for a Poisson(240) should overwhelmingly fall between 210 and 270, which is what your red plot shows. Try fitting a different distribution to your data.

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I just spotted StupidWolf's answer. Other than using a mean of 200 rather than 240, his histogram shows the same behavior described above.