When Does the Binomial Distribution Approximate the Poisson Distribution?

When Does the Binomial Distribution Approximate the Poisson Distribution?
In many statistical analyses, the binomial distribution and the Poisson distribution are commonly used to model various phenomena. Under certain conditions, the binomial distribution can be approximated by the Poisson distribution. This approximation is particularly useful in simplifying calculations and understanding underlying probabilities. Let's delve into the conditions under which this approximation is accurate and explore the limitations of this approximation.

Conditions for Approximation

The approximation of the binomial distribution by the Poisson distribution is valid under specific circumstances. The primary conditions are as follows:

Condition 1: Large Number of Trials
The number of trials, denoted by ( n ), must be sufficiently large. In the context of the binomial distribution, this means ( n to infty ).

Condition 2: Small Probability of Success
The probability of success ( p ) for each trial should be very small, such that ( p to 0 ).

Condition 3: Finite Expected Value
The product ( lambda np ) must remain finite and is often referred to as the parameter ( lambda ) of the Poisson distribution. This implies that the expected value of the number of successes remains constant.

Mathematically, if these conditions hold, the binomial probability can be approximated by the Poisson probability as follows:

[ Pr(X k) approx frac{lambda^k e^{-lambda}}{k!} ]

where ( lambda np ).

Illustrative Example in R

To see this in practice, consider an example using the R programming language. We'll use the `dbinom` function to calculate the probability of exactly 20 successes in 2200 trials with a success probability of 0.001, and compare it with the Poisson probability using the `dpois` function:

dbinom(20, size2200.001, prob0.001)[1] 0.0001866089lambda  0.02dpois(20, lambda)[1] 0.0001960397

Both probabilities are very close, indicating that the binomial distribution can indeed be approximated by the Poisson distribution under the given conditions.

Empirical Study and Limitations

Most textbook literature suggests that the Poisson distribution is a limiting case of the binomial distribution when ( n ) is large and ( p ) tends to zero. However, empirical studies sometimes reveal that the approximation can be accurate even when these conditions are not strictly met. In a study by Ramnath Takiar (2021), it was observed that even when ( n 10 ) and ( p 0.01 ), the Poisson probabilities are close to the corresponding Binomial probabilities. This is a significant finding as it challenges the conventional understanding that the Poisson approximation is only accurate when ( n ) is very large and ( p ) is very small.

The study by Takiar (2021) provides a more empirical approach to assessing the relationship between binomial and Poisson distributions. The results of this study strongly support the idea that the Poisson probabilities can be close to the binomial probabilities even when ( n ) is as low as 10 and ( p ) as high as 0.20.

Conclusion and Final Thoughts

The approximation of the binomial distribution by the Poisson distribution is a powerful tool in statistics, especially when dealing with large ( n ) and small ( p ). However, as evidenced by empirical studies, this approximation can be accurate under a wider range of conditions. It is crucial to understand the limitations and conditions under which such approximations hold true for more precise and reliable statistical analyses.

For those seeking to delve deeper into the relationship between the binomial and Poisson distributions, the reference to the study by Ramnath Takiar (2021) provides valuable insights and additional data to support this empirical approach.