In each of the situations, state whether the indicated model can be regarded as a Generalized Linear Model (GLM) and give reasons for your answer. 

Greek letters indicate unknown parameters, $Y_{i}$​ denotes the $i^{th}$ observation of a response variable correpsonding to the value $x_{i}$​ of an explanatory variable. Assume independent observations for i=1,...,n.

(i) $Y_{i}$​ =0 or 1 where P($Y_{i}=1$)= $e^{-\beta x_{i} }$
(ii) $Y_{i} \sim$ exponential with mean $\mu _{i}$ where $\mu _{i} = exp(\beta _{0} + \beta _{1}x _{i})$
(iii) $Y_{i} \sim  $​Poisson with mean $\mu _{i}$​ where $\mu _{i} = \eta _{i} e^{(\beta x _{i})}$ and $\eta _{i}$ ​ is known.
Commenting to add:

I believe that the answer for (ii) and (iii) are below and need my logic checked, for (i) I don't know how to approach this at all and need help with this.

For the others:

(ii) For exponential the mean given is $\mu _{i} = exp(\beta _{0} + \beta _{1}x _{i})$, thus if we can a log transformation of the mean, this will be a linear combination of XB and thus this is a GLM. Also note that the exponential distribution belongs to the exponential family.

(iii) Similarly, the mean here is $\mu _{i} = \eta _{i} e^{(\beta x _{i})}$, taking log this becomes log($\eta _{i} e^{(\beta x _{i})}$)=log($\eta _{i}) + {(\beta x _{i})}$ which is also a linear form of the predictor variable so this can also be regarded as a Generalized Linear Model. 


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Mathe Mathe
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