When looking at the shape of the histogram of a dataset, I always have difficulty discerning if I should model the data based on Poisson or Exponential distribution, as both seems reasonable.

Based on previous answers on CrossValidated, some have mentioned that one of the core tasks of assessing which distribution to use is looking at the relationship between mean and variance. However, in this case, both Poisson and Exponential distributions have mean equal to the variance.[Gordon Smyth pointed out exp dist has unequal mean and variance]

As such, when sample size is high (thousands), does it matter if my data is discrete or continuous? and how do I choose which of of the distributions I should use to model my data?

Can a discrete distribution be used on continuous data and vice versa?

edit: as Gordon Smyth pointed out, exponential distribution does not have equal mean and variance.

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    $\begingroup$ The Exponential distribution is a special case of the Gamma glm family, which has variance proportion to the square of the mean. The Poisson and Exponential distributions absolutely do not have the same mean-variance relationship. $\endgroup$ Commented May 12, 2020 at 2:34
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    $\begingroup$ IMO the answer to your question is no. Assume one is using glms, but I can't think of any reason to use a continuous distribution to model count data or discrete to model continuous. All glm distibutional families have different mean-variance relationships. Sample size does not affect the decision. $\endgroup$ Commented May 12, 2020 at 2:37
  • $\begingroup$ Thank you. And my apologies for stating exponential distribution has equal mean and variance. $\endgroup$ Commented May 12, 2020 at 3:22
  • $\begingroup$ Question might be easier to answer if you can give a specific application. Why is it important to give a name of the population distribution that yielded the sample plotted in your histogram. // The boundary btw discrete and continuous is often blurred. Continuous random variables are rounded, thus becoming discrete. Binomial and Poisson probabilties are sometimes approximated using Normal distribution. Etc. $\endgroup$
    – BruceET
    Commented May 12, 2020 at 4:33
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    $\begingroup$ @BruceET, thank you for the question! I received this question during an interview, where I was shown a histogram of discrete values (such as number of comments per user per month) and asked what distribution I would use the model it, and what is the mean and variance of this distribution (just the parameters, like lambda). I think they asked this as part of a hypothesis testing problem, where mean and std is used to calculate the effect size - given minimum detectable effect, find n needed for the test. The choice of distribution will affect the expected std of the new distribution. $\endgroup$ Commented May 12, 2020 at 4:58


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