W. B. Langdon
Computer Science, University College, London,
Gower Street, London, WC1E 6BT, UK
14 January 2001
Suppose we wish to maximise
subject to the conditions T>0, , and .
Since logarithm is a monotonically increasing function for the range of P of interest we can maximise (1) by maximising its logarithm. We do this by differentiating w.r.t. Pi for 1<i<T subject to the constraints. The maximum value of (1) occurs when each of the partial derivatives is zero.
Setting each partial derivative to zero (and noting ) yields
|-P1 + Pi||=||0|
That is the Pi when (1) is maximal are all equal. Since they still have to sum to C every Pi =1/C.