See my detailed response below. Don't forget to tip if you are satisfied.

The important thing you need to consider is that you are looking to capture important structures in the data, when you do statistical analysis and trying to forecast something using a data. The assumptions of normality and zero mean of residuals are important for statistical models, because they allow you to make valid statistical inferences/predictions.

When the residuals of a model are normally distributed and have a mean of zero, it indicates that the model is not leaving any systematic patterns or trends in the data. In other words, the model is able to capture the structure in the data, and the residuals represent the random variation that cannot be explained by the model. It indicates that the model is capturing most of the systematic patterns in the data. So, it is a good thing to have the residuals normally distributed with mean of zero. Otherwise, if the residuals have a non-zero mean or are not normally distributed, it suggests that there are systematic patterns or trends in the data that your model is not capturing, which could lead to incorrect inferences/predictions.

Regarding your question about wanting to see trends and systems in the data, it's important to note that the purpose of statistical modeling is not to eliminate all trends and systems in the data, but rather to capture the important ones while leaving behind the random variation as I mentioned above.

If there are systematic patterns or trends in the data that are relevant to your forecasting task, you would want to include them in your model. However, as described above, you would still want the residuals to be normally distributed with a mean of zero to ensure that the model is capturing the relevant structure in the data.

Regarding the use of random data to forecast, random data is not particularly useful for forecasting because it usually does not contain any structure or patterns that can be modeled and used for prediction. But, it depends! (See below too). In forecasting, you typically want to capture the underlying structure in the data that allows you to make predictions about future values. Random data, by definition, does not contain any underlying structure or patterns that can be used for forecasting.

I should note that randomness is not always a bad thing in data. The absence of any systematic patterns or trends (i.e. randomness) can be useful in certain cases. For example, if you are trying to test a hypothesis that assumes the data is random, then having systematic patterns in the data could lead to incorrect conclusions.

Randomness in data can also be useful in certain cases for forecasting. For instance, if you are trying to forecast demand for a new product or service (e.g. forecasting demand for a new iPhone, or a new electric car), having random data can help you estimate the uncertainty around your forecast. Randomness can also help you identify new patterns or trends that were not present in your historical data.

To use random data for forecasting, you can use time series models that explicitly model the randomness in the data. For example, you can use an ARIMA model to forecast a time series with random data.

Low bounty!

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