630 BELL SYSTEM TECHNICAL JOURNAL 



Similarly in n dimensions, suppose the second order moments of 

 p(xi , ' ■ ■ , Xn) are fixed at A ij : 



Aij = I • • • j XiXjpixi , ■ ' ■ ,Xn) dXi '" dXn. 



Then the maximum entropy occurs (by a similar calculation) when 

 p{xi , ' ' ' , Xn) is the n dimensional gaussian distribution with the second 

 order moments A ij . 



6. The entropy of a one-dimensional gaussian distribution whose standard 

 deviation is cr is given by 



Hix) = log \/2ire(T. 



This is calculated as follows: , 



-log p{x) = log V27ro- + ;^ 

 H{x) = — j p(x) log p{x) dx 



= j p(x) log \/2^adx + j p{x) 1^2 d^ 



2 



= log V2i(T + ^2 



= log \/2ir(r + log \/e 



= log -s/lirea. 



Similarly the n dimensional gaussian distribution with associated 

 quadratic form aij is given by 



p{x,,---, x„) = I0i^ exp (- JSayJ^.X,) 



and the entropy can be calculated as 



ff = log(2«r'^|a«|' 



where | aij \ is the determinant whose elements are aij . 



7. If X is limited to a half line (p(x) = for x < 0) and the first moment of 

 X is fixed at a: 



= / p{x)x dXj 

 Jo 



