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”)**

**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.

## Be the first to comment on "Poisson Distribution as a Limiting Case of Binomial Distribution"