Using your Head is Permitted

November 2012 solution

A standard normal random variable has probability density proportional to exp(-x2/2) to attain the value x. To get all n random variables to specific values together, the probability density is proportional to exp(-(x12+...+xn2)/2). If we think of (x1,...,xn) as a vector in Euclidean space, this can be put in terms of its length, r, as exp(-r2/2). 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 S2/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 S2/n+Χ2(n-1).

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