My question is about the nomenclature of specifying models. I have a specific example, taken from a paper:
The outcome $Y$ is an independent Poisson variable, with means determined by the parameters $\alpha$ and $\beta$, such that $$ Y \sim \text{Poisson}(\alpha\beta)$$
Q1) What is the meaning of this statement? From what I understand, the distribution of Poisson variable is given by with mean equal to $\lambda$. Thus, from the specification above, is $\lambda = \alpha\beta$?
Q2) if I wanted $\lambda = \alpha + \beta$, would the model specification then be:
$$ Y \sim \text{Poisson}(\alpha + \beta)$$
Q3) How would one specify logistic regression models and linear models in this manner? Particularly for logistic models I have seen
$$\text{logit}(\text{Pr}(Y = 1)) = \beta_0 + \beta_1 x_1 + \dotsm$$
How is this converted into the specification format given above?