How the Poisson distribution is used in regression? I am trying to intuitively understand the Poisson regressions, but I don't get:

*

*why the increasing variance along with the mean is relevant?

*how each of these three theoretical Poisson distributions was shown here, i.e. using each Y value as a mean=variance?

So far, my understanding is that for each Y a theoretical distribution is drawn, and then the best curve is fitted to maximize the cumulative probabilities (maximum likelihood?)



why the increasing variance along with the mean is relevant?

Remember that the variance of a Poisson random variable is equal to its mean.  If the mean increases, so too does the variance.

how each of these three theoretical Poisson distributions was shown here, i.e. using each Y value as a mean=variance?

Regression is all about the conditional distribution of the outcome.  If you fix $x$ and observe $y$ a bunch of times, and then look at the histogram of those $y$, then that histogram should be Poisson.
