Using your Head is Permitted

November 2012 riddle

Let X1,..., Xn be independent, standard normally distributed random variables.

Let S be the sum of all Xi.

Let Q be the sum of all Xi2.

It is well known that S is distributed normally, with mean 0 and variance n. It is also well known that Q is distributed chi-squared with n degrees of freedom.

This month's challenge: what is the distribution of Q given S?

A big thank-you goes this month to Omer Angel, who showed me the elegant solution.

List of solvers:

Jan Fricke (2 November 06:37)
Ganesh Lakshminarayana (10 November 03:21)
Jin Ruizhang (12 November 13:29)
Albert Stadler (23 November 21:45)

Elegant and original solutions can be submitted to the puzzlemaster at Names of solvers will be posted on this page. Notify if you don't want your name to be mentioned.

The solution will be published at the end of the month.


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