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Relationship between poisson and exponential distribution Note, that a poisson distribution does not automatically imply an exponential pdf for waiting times between events This only accounts for situations in which you know that a poisson process is at work But you'd need to prove the existence of the poisson distribution AND the existence of an exponential pdf to show that a poisson process is a suitable model!
When to use an offset in a Poisson regression? [duplicate] Here is an example of application Poisson regression is typically used to model count data But, sometimes, it is more relevant to model rates instead of counts This is relevant when, e g , individuals are not followed the same amount of time For example, six cases over 1 year should not amount to the same as six cases over 10 years So, instead of having $$\log \mu_x = \beta_0 + \beta_1 x
What is the appropriate model for underdispersed count data? The Poisson and the negative binomial distribution assume a higher dispersion, so that shouldn't be appropriate, right? I was thinking about applying the quasi-Poisson distribution, but that's usually recommended for over-dispersion
Conceptual difference between Poisson and uniform distribution The Poisson distribution is indeed a discrete distribution, but I think your confusion seems to follow from the fact that what you describe are the properties of the Poisson process, not the distribution