# Poisson approximation to the normal

• rhyno89
In summary, the central limit theorem states that for values of lambda greater than 10, but ideally greater than 32, the Berry-Esseen theorem can be used to find the difference between the CDF of a sample mean and the normal CDF. By writing a high-frequency Poisson as a sum of low-frequency iid Poissons, the value of n can be found to achieve the desired accuracy.
rhyno89
So my book merely mentions that this holds as a result of the central limit theorem for values of lambda greater than 10, but ideally greater than 32.

Anyway I was wondering if anyone knew this actual proof as I am interested in seeing it step by step and I could not have found it anywhere that I have looked.

Thanks

I can't give you an off hand answer, but it is essentially based on the central limit theorem. Similar result holds for binomial distribution.

Last edited:
The Berry-Esseen theorem is similar to CLT but gives a bound on the difference between the CDF of a sample mean and the normal CDF, in terms of n and the third moment.

To use this theorem here, for example, write a high-frequency Poisson as a sum of low-frequency iid Poissons (e.g. $$\lambda=n.f$$ where f is a value between 0.9 and 1) and then find the value of n that gives the required accuracy.

## 1. What is the Poisson approximation to the normal distribution?

The Poisson approximation to the normal distribution is a mathematical approximation used to estimate the probability of rare events occurring. It is based on the assumption that the Poisson distribution, which models the occurrence of rare events, can be closely approximated by the normal distribution, which is commonly used to model continuous data.

## 2. How is the Poisson approximation to the normal distribution calculated?

The Poisson approximation to the normal distribution is calculated using the formula P(X = k) = (λ^k * e^(-λ)) / k!, where λ is the mean of the Poisson distribution and k is the value of interest.

## 3. What are the assumptions of the Poisson approximation to the normal distribution?

The main assumption of the Poisson approximation to the normal distribution is that the events being modeled are rare, meaning that their probability of occurrence is small. Additionally, it assumes that the events are independent and that the sample size is large enough to apply the central limit theorem.

## 4. When is the Poisson approximation to the normal distribution used?

The Poisson approximation to the normal distribution is commonly used in situations where we are interested in estimating the probability of rare events occurring. This can include areas such as finance, insurance, and quality control, where rare events can have significant consequences.

## 5. What are the limitations of the Poisson approximation to the normal distribution?

While the Poisson approximation to the normal distribution can be a useful tool, it is important to note that it is an approximation and may not always provide accurate results. It is also limited to rare events and may not be appropriate for data that follows a different distribution or for events that occur frequently.

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