I have some data from a lake. "Counts" is the response variable and is a number for the organism in the lake. I also have a variable "stations" that indicate every parts of the lake (there are 45 stations).
It can be assumed that: $$Y_i=\mu_i+\epsilon_i$$ with $E\epsilon_i=0$, $V \epsilon_i=\psi V(\mu_i)$
and $\mu_i=\beta_{stations}$ being a station specific station mean value. It will makes to use normal distribution as response distribution.
Then I be introduced to a new law the Taylor Law: It states that there is a power law relationship between mean and variance when counting organism. And specifically the variance function is given by: $$V(\mu)=\mu^b$$ for some power $b≥0$. And $\xi_i=E\epsilon_i^2$
Then I have to show that if the Taylors law holds then $$log(\xi_i)=log(\psi)+blog(\mu_i)$$
And I have to show that if $\epsilon_i$ is normally distributed then $V \epsilon_i^2=2 \xi_i^2$.
I understand it most, intuitively. The $\epsilon$ is the residual error and $\mu$ is the mean and then there is a relationship between the mean and variance for some unknown $b≥0$. But I'm a bit confused what $\xi$ and $\psi$ is and and how I should gather the information to show these two statements. I hope anyone can help me?