# Using your Head is Permitted

## November 2012 solution

A standard normal random variable has probability density proportional to
exp(-*x*^{2}/2) to attain the value *x*. To get all *n*
random
variables to specific values together, the probability density
is proportional to
exp(-(*x*_{1}^{2}+...+*x*_{n}^{2})/2). If we
think of (*x*_{1},...,*x*_{n}) as a vector in
Euclidean space, this can be put in terms of its length, *r*, as
exp(-*r*^{2}/2). Importantly, the distribution is invariant to
rotation around the origin.
Because of this property, we can rotate our frame of reference: switch to a new
set of variables, by using coordinates along a different orthogonal system.
Just as before, the new variables are independent and individually standard
normal.

If one of the base vectors of the new system is
(1/√*n*,...,1/√*n* ), then constraining *S* is merely
constraining this variable, with no effect on any of the other coordinates.
Specifically, the variable will take the value *S*/√*n* . Its
square is *S*^{2}/*n*. The rest of the variables remain
standard normal and independent, but now there are only *n*-1 of them.

In total, the distribution of *Q* is
*S*^{2}/*n*+Χ^{2}(*n*-1).

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