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?

E

Expert

Verified

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.

   Related Questions in Financial Management

©TutorsGlobe All rights reserved 2022-2023.