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What I learned from book is predicting counting is different from predicting a continuous variable. For example, if we want to predict how many mails a person get per day, we may use Poisson regression.

However, if the counting number is big (for example, if we want to predict how many days are sunny in a year.), can we just treat it as a continuous variable? When $\lambda$ is large, the we can use normal distribution to approximate Poisson distribution. Am I right?

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A random variable is a counting variable does not only mean that it has natural number values. So, the number of sunny days in a year is not a counting random variable, because it is not the result of a counting process. Probably, a day is declared sunny if some burocratic criteria are fulfilled, like at least 5 hours clear sun, or whatever. It is not a count of independent events. Examples of count data is: number of auto accidents in new york, per day. Number of stillbirths in Guatemala, per day. These are counting independent events, which could as a first approximation be modelled via Poisson distribution or a poisson point process. I can see no such Poisson model lurking behind number of sunny days! For instance, have a look at my answer here: Goodness of fit and which model to choose linear regression or Poisson The arguments used there are irrelevant in your case.

Back to your question, if "the count is big". It is not the bigness in itself that matters, big counts could still be Poisson (but big counts would in practice often be clustered and some more complicated model than poisson would be needed). For number of sunny days in a year, you could sure try ordinary linear regression, as a starting point.

To elaborate on why "number of sunny days" is not a count variable. First, number of hours of (sufficiently strong) sunshine is measured at meteorological stations with a Campbell–Stokes recorder, see https://en.wikipedia.org/wiki/Campbell%E2%80%93Stokes_recorder They look like this:

image of a Campbell-Stokes recorder

and works by focusing the sun on a paper clip, and burning a path there when the sun is strong enough. Then one have to measure the length of the burnt path. That gives a measured variable, not a count variable! The underlying process is measurement, not counting. Then this is converted into a binary sunny/not sunny indicator by some arbitrary ("burocratic") cutoff. Hope this is a better explanation of my answer!

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    $\begingroup$ I believe your characterization of "count" data is much more restrictive than most people intend. Count data are special not because they are assumed to have Poisson distributions! They can indeed be dependent, too. Whether or not the criteria for determining what is counted are "bureaucratic" or not doesn't matter, either. The number of sunny days is an excellent example of count data. $\endgroup$
    – whuber
    Dec 2, 2016 at 16:25
  • $\begingroup$ @whuber: I admit I have never seen a formal definition of "count data", but if it is'nt based on some kind of point process (yes, independence is to strong) then the concept becomes to opaque to have much meaning. What do counting number of sunny days have in common with counting accidents? (except of the word "counting") $\endgroup$ Dec 2, 2016 at 16:30
  • $\begingroup$ @kjetilbhalvorsen thanks for the knowledge about sunshine measure!, I did not expect I will get this level of detail. Sunny days is just the first example in my head. So, according to your answer, it really matters on how the variables are measured? and one would different than another? How about something like, number of meetings a person attend in a month or number of active days in the online form? It would be different than sunny days analogy? $\endgroup$
    – Haitao Du
    Dec 2, 2016 at 16:59
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    $\begingroup$ Measurement is a red herring here. When you're counting green jelly beans, you are making a measurement (of color). When you're counting isopods, you are classifying them into a genus or size class according to measurements. When you're counting sunny days, you are classifying them into sunny and not-sunny by making measurements. The fact that the measurement might be continuous, or subject to uncertainty, or correlated, or even arbitrary, is irrelevant to the fact that you're dealing with counted data. $\endgroup$
    – whuber
    Dec 2, 2016 at 18:28
  • $\begingroup$ This is being argumentative, the underlying problem being that there is no (or I can't find any) "official" definition of "count data" or "count data models". I have been googling a little, did not find a single paper or website that bothers to define "count data". But taking the sunshine example, if you are "counting", then you are counting successes (or failures), not events. That lead to a binomial distribution, maybe logistic regression, which is noy usually included in treatments of "count data models". But without definitions, no surprise that we start arguing ... $\endgroup$ Dec 2, 2016 at 18:44
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At the fundamental level of chemistry and atomic theory, one could argue that the world is discrete rather than continuous. One might argue that continuous variables themselves are just very useful approximations to an underlying discrete reality. So clearly it is OK to treat counts as continuous variables. We do it all the time in practice.

This is different from the issue of whether Poisson approximations are appropriate for any particular application. The answer from @kjetil covers that well.

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