## Info

n = 5 and this is to be expected since there are twice as many data points in the former case. But the curve marked n = 10 is also steeper than that marked n = 5: it rises from our best estimate of approximately 6 in a sharper manner than the curve marked n = 5. Based on our experience with likelihood, and Hudson's formula (Eq. (3.32)), we would expect that the steepness should tell us something about the likelihood of different values of m.

To answer that question, that is to be able to assess the likelihood of different values of m rather than just find the best estimate, we need to introduce a statistical model. And what would the simplest model be? How about this one:

where X is N(0, 1). If we accept Eq. (3.75) as the statistical model, then the sum of squared deviations is the same as ^¿L1 (X/)2.

The sum of the squares of n normally distributed random variables is a new statistical quantity for us. It is called the chi-square distribution with n degrees of freedom and has probability distribution function

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