Poisson processes and the Poisson distribution

Statistical Thinking in Python (Part 1)

Justin Bois

Teaching Professor at the California Institute of Technology

Poisson process

  • The timing of the next event is completely independent of when the previous event happened
Statistical Thinking in Python (Part 1)

Examples of Poisson processes

  • Natural births in a given hospital
  • Hits on a website during a given hour
  • Meteor strikes
  • Molecular collisions in a gas
  • Aviation incidents
  • Buses in Poissonville
Statistical Thinking in Python (Part 1)

Poisson distribution

  • The number r of arrivals of a Poisson process in a given time interval with average rate of lambda arrivals per interval is Poisson distributed.
  • The number r of hits on a website in one hour with an average hit rate of 6 hits per hour is Poisson distributed.
Statistical Thinking in Python (Part 1)

Poisson PMF

ch3-4.016.png

Statistical Thinking in Python (Part 1)

Poisson Distribution

  • Limit of the Binomial distribution for low probability of success and large number of trials.
  • That is, for rare events.
Statistical Thinking in Python (Part 1)

The Poisson CDF

samples = rng.poisson(6, size=10000)

x, y = ecdf(samples)
_ = plt.plot(x, y, marker='.', linestyle='none')
plt.margins(0.02)
_ = plt.xlabel('number of successes')
_ = plt.ylabel('CDF')
plt.show()
Statistical Thinking in Python (Part 1)

The Poisson CDF

ch3-4.023.png

Statistical Thinking in Python (Part 1)

Let's practice!

Statistical Thinking in Python (Part 1)

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