Review of (General) Linear Model

*Generalized Linear Models

Linear Regression as GLM

Normal Distribution as an Exponential Family

Recall:

$$ f(y|\theta,\phi)=exp\{\frac{y\theta-b(\theta)}{\phi}+c(y,\phi)\} $$

For Normal Distribution

$$ f(y|\mu,\sigma^2)=exp\{\frac{\mu y-\frac{1}2\mu^2}{\sigma^2}-\frac{y^2}{2\sigma^2}-\frac{1}2ln(2\pi\sigma^2)\} $$

where

• $\theta=\mu$
• $\phi=\sigma^2$
• $b(\theta)=\frac{1}2\mu^2$
• $c(y,\phi)=-\frac{y^2}{2\sigma^2}-\frac{1}2ln(2\pi\sigma^2)$

*(All the 3 equations above are equivalent)

And

• $E(y)=b'(\theta)=\mu$
• $var(y)=b''(\theta)\phi=\sigma^2$

Moments