**RN/01/14**

**
W. B. Langdon
W.Langdon@cs.ucl.ac.uk
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. *P*_{i} 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

= | 0 | ||

= | 0 | ||

-P_{1}
+
P_{i} |
= | 0 | |

P_{i} |
= | P_{1} |

That is the *P*_{i} when (1) is maximal are all equal.
Since they still have to sum to C
every *P*_{i} =1/*C*.