I am thinking about the relationship between sample mean and variance in an example. If we want to look at the average goals per month for a soccer team. And we have mean and variance of goals for each month. Now we find average goals per match is higher if goals per match in a month variates a lot. Is the high correlation between mean and variance defined by math, or it deserved to explore.
It may indicate that some part of the team is impacting the number of goals. For example the strategy, the team may have a rotation on players. So substitutes have no goals in first 2 matches, and starting lineup have extremely high goals after well rest. In this case, mean would be greater than having starting 11 players on every match and getting really tired.
Further, if we add samples variance to the nonlinear model to predict mean, dose the variance provides a unique contribution to the model, or it is more of a self-learning?