# Poisson distribution, Standard deviation, fitting line

Let's say I have system A, B, C, and D. Each system contains 10,000 numbers generated by Poisson distribution. The difference is the mean is different for different systems. I calculated the std dev for each distribution corresponding to each system. I plot the standard deviation with respect to mean numbers. (I observe std deviation increases with mean). I want to fit the plot with some line, which gives me the general estimate of std dev with respect to the mean number. I need help in finding the fitting line!

Thank you!

Maybe I misunderstand the question, but the standard deviation of the Poisson distribution is the square root of the mean. So that's the exact solution right away.

If you fit y = X B you're trying to fit a straight line to sqrt(x). It's not going to fit very well. Try to fit a line to the squares of the standard deviations instead, and you'll find B = 1 meaning y = x.

This sounds like simple linear regression. Your x-value is the vector of means and the y-value is the vector of corresponding standard deviations:

$$Y = X\beta$$

$$\beta$$ a vector containing the slope of your regression line and the offset term.

In R for example you can do fit <- lm(Y~X).