Understanding Gaussian Multiplication (Chapter 4)

[yehudatager]

Perhaps a more reasonable assumption would be that one person made a mistake, and the true distance is either 10.2 or 9.7, but certainly not 9.95. Surely that is possible. But we know we have noisy measurements, so we have no reason to think one of the measurements has no noise, or that one person made a gross error that allows us to discard their measurement. Given all available information, the best estimate must be 9.95."

I was thinking about it, and I think that this can be explained mathematically. Imagine the standard deviation is 0.25. If the real value is 9.95, then each reading is one standard deviation off (in opposite directions). The odds of that are 16%. However, if the real value is 10.2, then the other sensor is two standard deviations of, a probability of 5%. A similar argument can be made regardless of the value of the standard deviation.

In real life, this does not always make sense. The odds of one measurement being correct and the other being way off might be higher than the odds of both being significantly off. However, real life is not Gaussian.

Thoughts?