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Determine standard deviation and probability

You have one hat containing normally distributed random numbers, with a mean of zero and a standard deviation of σ which is unknown. You draw N numbers φi from this hat. What is the ‘probability’ of drawing all of the numbers φ1, φ2, ... , φN from independent Normal distributions with mean zero and standard deviation σ?

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1524_volatility1.png

Now choose the σ which maximizes this quantity. This is easy. First take logarithms of this expression, and then differentiate with respect to σ and set result equal to zero:

804_volatility2.png

(A multiplicative factor has been avoided here.) It is:

1909_volatility3.png

Thus, our best guess for σ is specified by

982_volatility4.png

You must recognize it as a measure of the variance.

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