Lets say I want to generate 10,000 random samples from a normal distribution. Is there anything wrong with first generating 10,000 numbers, then computing the mean (which will be a value close to 0) of the population, and finally subtracting this mean from each sample? Doing so gets me 10,000 samples that truly do have a mean of 0.
Further, could we do a similar "correction" for the variance? We could do this by dividing each sample by the standard deviation of the population. After doing this correction, we could then correct the mean as described above.
These procedures will give us a population that truly has a mean of 0 and variance of 1, which is what we wanted. However, I'm not sure if these procedures introduce problems that I am not aware of?
If your numbers are independent normal distributed random numbers, that’s as good as it gets. If you “correct” a sequence, it’s not random anymore. It can easily be distinguished from a random sequence - because the average is too close to 0.