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Poisson Distribution

Poisson Distribution is a Discrete Distribution.

It estimates how many times an event can happen in a specified time. e.g. If someone eats twice a day what is the probability he will eat thrice?

It has two parameters:

  • lam- rate or known number of occurrences e.g. 2 for above problem.
  • size- The shape of the returned array.

Example

Generate a random 1x10 distribution for occurrence 2:

from numpy import random

x = random.poisson(lam=2, size=10)

print(x)

Visualization of Poisson Distribution

Example

from numpy import random
import matplotlib.pyplot as plt
import seaborn as sns

sns.distplot(random.poisson(lam=2, size=1000), kde=False)

plt.show()

Result

02468

Difference Between Normal and Poisson Distribution

Normal distribution is continuous whereas poisson is discrete.

But we can see that similar to binomial for a large enough poisson distribution it will become similar to normal distribution with certain std dev and mean.

from numpy import random
import matplotlib.pyplot as plt
import seaborn as sns

data = {
  "normal": random.normal(loc=50, scale=7, size=1000),
  "poisson": random.poisson(lam=50, size=1000)
}

sns.distplot(data, kind="kde")

plt.show()

Result:

bionominal distribution

Difference Between Binomial and Poisson Distribution

Binomial distribution only has two possible outcomes, whereas poisson distribution can have unlimited possible outcomes.

But for very large n and near-zero p binomial distribution is near identical to poisson distribution such that n * p is nearly equal to lam.

from numpy import random
import matplotlib.pyplot as plt
import seaborn as sns

data = {
  "binomial": random.binomial(n=1000, p=0.01, size=1000),
  "poisson": random.poisson(lam=10, size=1000)
}

sns.distplot(data, kind="kde")

plt.show()

Result:

bionominal distribution