# 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

Probability distribution of Poisson random variable is

## 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

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

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.