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Why is Poisson regression used for count data? Poisson distributed data is intrinsically integer-valued, which makes sense for count data Ordinary Least Squares (OLS, which you call "linear regression") assumes that true values are normally distributed around the expected value and can take any real value, positive or negative, integer or fractional, whatever Finally, logistic regression only works for data that is 0-1-valued (TRUE-FALSE
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!
Why Specifically Use Poisson Regression For Count Data? Why should Poisson Regression be used for Count Data instead of a "vanilla linear regression"? I understand the basic argument : Count Data is by definition discrete and you would rather use a model in which predictions are always discrete (i e Poisson Regression) but to me, this seems like a formality
r - Rule of thumb for deciding between Poisson and negative binomal . . . The Poisson distribution implies so a one-sample test can provide a -value for testing Poisson vs negative binomial Another test for equidispersion is the Lagrange Multiplier which follows a one-degree distribution under the null
Finding the probability of time between two events for a poisson process The logic here seems obvious: The probability of a given wait time for independent events following a poisson process is determined by the exponential probability distribution $\lambda e^ {-\lambda x}$ with $\lambda = 0 556$ (determined above), so the area under this density curve (the cumulative probability) is 1