Poisson Distribution as a Limiting Case of Binomial Distribution

For large value of n binomial distribution asymptotically tends to Poisson distribution.

Probability distribution  function of binomial random variable  is

PDF of Binomial Distribution
Probability distribution of Poisson random variable is

 Poisson Distribution PDF

Poisson Distribution as a Limiting Case of Binomial Distribution

Poisson Distribution as a Limiting Case of Binomial Distribution

Python Code for Binomial Distribution


from scipy.stats import binom
import numpy as np
import matplotlib.pyplot as plt
# Let lambda=np=5
x = np.arange(0,10)
n=50
p=0.10
plt.plot(x, binom.pmf(x, n, p))
plt.savefig(“binom.png”)


For n=50 and p=0.10  and λ=5   x=0…..10

The plot is

Binomial Distribution For Small n

For n=1000 and p=0.005  where λ=5 and    x=0…..100


from scipy.stats import binom
import numpy as np
import matplotlib.pyplot as plt
# Let lambda=np=5
x =np.arange(0,100)
n=1000
p=0.005
plt.plot(x, binom.pmf(x, n, p))
plt.savefig(“binom.png”)


Output would be
Binomial Distribution for Large n

Conclusion

In  this post, I have derived probability distribution function of Poisson distribution using Stirling formula. Further, using Stirling formula many distributions can be derived in limiting case. Moreover, I hope you will also derive another distributions.

 

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