# Probability distribution & Interpretation

If my distribution looks similar to Poisson but not an actual Poisson distribution which is verified using a QQ plot, is there any way to convert such distribution to a proper Poisson distribution using python? If yes, would the same method work for binomial distribution and other discrete geometric and continuous distributions?

Question 2:

When empirical distributions like Poisson distribution has their own has a set of descriptors, why is it a general practice to convert all the unknown distributions to gaussian.

• What is the problem you are trying to solve? Why do you bother about it being a Poisson distribution?
– Tim
Commented Jan 25, 2022 at 7:59
• My intention is to know, why so much obsession over gaussian, and why not other distribution. Then, I know we can use box-cox transform to convert any unknown distribution to normal, is there any other transform to convert a Poisson-like distribution to an actual Poisson distribution Commented Jan 25, 2022 at 8:18
• Mathematically, the transform of $X\sim F$ into $Y\sim G$ can always be obtained by $$Y=G^{-1}(F(X))$$ Commented Jan 25, 2022 at 9:14
• That works if F and G are continuous; but this doesn't break up a big spike of probability into smaller ones (you can't convert a geometric with mean 0.25 into a Poisson with mean 10, for example) Commented Jan 25, 2022 at 16:07

## 1 Answer

If my distribution looks similar to Poisson but not an actual Poisson distribution which is verified using a QQ plot,

We use the probability distributions to approximate the distribution of the data. Empirical distribution won't exactly fit the theoretical distribution.

is there any way to convert such distribution to a proper Poisson distribution using python?

Poisson distribution is a distribution over non-negative integers, with mean equal to variance. As noticed in comment by @Xi'an, technically you could use inverse transform. But why would you want to do that?

why is it a general practice to convert all the unknown distributions to gaussian.

It isn't. We sometimes transform the data using things such as log transformations, or Box-Cox. There are many reasons for and against transformations in different cases. As noticed by Nick Cox:

The general idea of transformation is that it can be easier to see and analyze what is happening on a transformed scale, while specifically there are many techniques for which some approximation to normal distribution(s) provides, if not conditions that are assumed to be true, as so often stated, then at least relatively ideal conditions for summary and inference.

TL;DR we don't "always" transform the variables. We do it sometimes, to make out life easier. We don't expect the data to look exactly like Poisson or normal distribution, we use them only as approximations. You shouldn't be doing things like reshaping the data to fit the desired distribution without good reason, at best this would make your results hard to interpret, if not hard to justify.