L21.10 The Poisson Approximation to the Binomial

TL;DR

In situations with a large number of slots but a small probability of success in each slot, the binomial distribution can be approximated with the Poisson distribution.

Transcript

We have seen that the binomial distribution plays an important role in the study of the Bernoulli process. And the reason is that the binomial distribution describes the number of arrivals during a fixed number of slots. We will now develop an approximation to the binomial distribution that applies to one particular regime, and that regime is when ... Read More

Key Insights

• #️⃣ The binomial distribution describes the number of arrivals during a fixed number of slots in the Bernoulli process.
• 🦥 An approximation to the binomial distribution can be developed in situations with a large number of slots and a small probability of success in each slot.
• ⛔ The Poisson distribution can be used as an approximation for the binomial distribution in the given limit.
• ⌛ This approximation is useful in situations such as studying earthquakes over a long time frame or dividing time into small slots.

Questions & Answers

Q: What role does the binomial distribution play in the study of the Bernoulli process?

The binomial distribution describes the number of arrivals during a fixed number of slots in the Bernoulli process.

Q: When can the binomial distribution be approximated with the Poisson distribution?

The binomial distribution can be approximated with the Poisson distribution in situations with a large number of slots and a small probability of success in each slot.

Q: Can you provide an example of a situation where this approximation might arise?

For example, if you're interested in earthquakes over a long time frame and divide time into small slots, you would have a large number of slots but a small probability of having a noticeable earthquake in each slot.

Q: What is the formula for the Poisson PMF that approximates the binomial PMF?

The formula for the Poisson PMF, which approximates the binomial PMF in the given limit, is e^(-λ) * (λ^k) / k!

Summary & Key Takeaways

• The binomial distribution is important in the study of the Bernoulli process, which describes the number of arrivals during a fixed number of slots.

• An approximation to the binomial distribution can be developed when there are a large number of slots and a small probability of success in each slot.

• In situations like earthquakes over a long time frame or dividing time into very small slots, the binomial distribution can be approximated with the Poisson distribution.