function val = min_std_dev_fn(wj, pj,qj)
% MIN_STD_DEV_FN - Returns the standard deviation of the horse track betting problem
%
% wj has N-1 elments the last element wj(N) is computed from the previous N-1 i.e. = 1 - sum( wj )
%   
% Written by:
% -- 
% John L. Weatherwax                2006-12-31
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

% "create" the probabilty we must assign to the last horse:
wj = [ wj(:); 1-sum(wj) ]; wj = wj(:)'; 

% compute the mean
%
m = sum( pj .* wj .* ( qj + 1 ) ) - 1;

% compute the  standard deviation: 
%
val = sqrt( sum( pj .* ( wj .* ( qj + 1 ) - 1 - m ).^2 ) );