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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?

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Then I have to show that if the Taylors law holds then $$log(\xi_i)=log(\psi)+blog(\mu_i)$$

You have this equation

$$V\epsilon_i = \psi V(\mu_i) $$

and since $V(\mu_i) = \mu_i^b$ you get

$$V\epsilon_i = \psi \mu_i^b$$

if you take the logarithm

$$\log V\epsilon_i = \log \psi + b \, \log \mu_i$$

which is the equation that follows from Taylor's law. And $V\epsilon_i = E[\epsilon_i^2] = \xi_i $ since $E[\epsilon_i] = 0$ by assumption.

But I'm a bit confused what $\xi$ and $\psi$

The variable $\xi_i$ is equal to $E[\epsilon_i^2]$. The variable $\psi$ is this parameter in your first equation.

And I have to show that if $\epsilon_i$ is normally distributed then $V \epsilon_i^2=2 \xi_i^2$.

All these Greek letters make it complicating. What is asked is to show that if $X$ is normal distributed (and with mean zero), then what is the variance of $X$? Hint: the distribution of $X$ is a gamma distribution (or can be seen as a scaled chi-squared distribution).

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