# Approximation of skellam distribution by a Gaussian one

• sabbagh80
In summary, the conversation discusses the use of a Gaussian distribution as an approximation for the Skellam distribution in order to find the probability of n being greater than or equal to 0. It is noted that the Gaussian distribution is a good approximation when at least one of the lambdas is large, but not always the case. The concept of maximum likelihood is also mentioned as a way to approximate one distribution with another.
sabbagh80
Hi, everybody

Let $n_1$ ~ Poisson ($\lambda_1$) and $n_2$ ~ Poisson ($\lambda_2$).
Now define $n=n_1-n_2$. We know $n$ has "Skellam distribution" with mean $\lambda_1-\lambda_2$ and variance $\lambda_1+\lambda_2$, which is not easy to deal with.
I want to find the $Pr(n \geq 0)$. Is it possible to find a good approximation for the above probability by employing an approximated "Gaussian distribution"? If "Gaussian" is not a good candidate, which distribution can I replace it with?

If at least one of the lambdas is large, the Gaussian with the same mean and variance will be a good approximation.

But it is not always the case. I want to deal with the more general cases.

Yes, I know. But for small lambda, I don't think there's any simpler approximation. Of course, you could in that case just truncate the distributions. If the mean is small, the probably of large n is vanishingly small, so it won't introduce much inaccuracy to leave them out.

To approximate one distribution with another use maximum likelihood, i.e. maximize
$$E[\log(f(X;t))$$
wrt the parameter vector t, where f is the pdf or pmf of the approximating distribution. E.g. solving for the normal distribution we get $\mu=E[X]$ and $\sigma^2=E[X^2]-E[X]^2$.

bpet said:
To approximate one distribution with another use maximum likelihood, i.e. maximize
$$E[\log(f(X;t))]$$
wrt the parameter vector t, where f is the pdf or pmf of the approximating distribution. E.g. solving for the normal distribution we get $\mu=E[X]$ and $\sigma^2=E[X^2]-E[X]^2$.

Could you please explain it in more details.

## 1. What is the Skellam distribution?

The Skellam distribution is a discrete probability distribution that is used to model the difference between two independent counts of events. It is often used in sports analytics to model the difference in scores between two teams or in biology to model the difference in gene expression between two groups.

## 2. What is the Gaussian distribution?

The Gaussian distribution, also known as the normal distribution, is a continuous probability distribution that is commonly used in statistics. It is often used to model real-world phenomena such as heights, weights, and test scores. It is characterized by a bell-shaped curve and is widely used due to its mathematical simplicity and numerous applications.

## 3. Why is there a need to approximate the Skellam distribution with a Gaussian one?

The Skellam distribution can be difficult to work with due to its discrete nature, making it challenging to calculate probabilities and perform statistical analyses. On the other hand, the Gaussian distribution is continuous and has well-known properties, making it easier to work with mathematically. Therefore, approximating the Skellam distribution with a Gaussian one allows for simpler and more efficient computations.

## 4. How is the Skellam distribution approximated by a Gaussian one?

The Skellam distribution can be approximated by a Gaussian distribution using a technique called the central limit theorem. This theorem states that the sum of a large number of independent and identically distributed random variables will be approximately normally distributed. Therefore, by considering the Skellam distribution as the sum of many independent Poisson distributions, we can approximate it with a Gaussian one.

## 5. What are the limitations of approximating the Skellam distribution with a Gaussian one?

While approximating the Skellam distribution with a Gaussian one can make calculations easier, it is important to note that this is only an approximation. The accuracy of the approximation depends on the number of events being modeled and the difference between the means of the two Poisson distributions. Therefore, for small sample sizes or large differences between means, the approximation may not be accurate.

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