q3b1.m

function [E, V] = q3b1(E_Rh, E_Rb, E_Rs, V_Rh, V_Rb, V_Rs, HB, HS, BS)
% Computes the expected value and variance of the total loss of points.
%
% Input :
%   * E_Rh, E_Rb, E_Rs : Marginal expected vaues.
%   * V_Rh, V_Rb, V_Rs : Marginal variances.
%   * HB, HS, BS       : Joint laws probabilities.
%
% Output :
%   * E    : Total points lost expected value.
%   * V    : Total points lost variance.

% R_h = g_h(H). Random Variable with the points reduced for each H.
g_h = [0; -2; -3; -5; -7];
% R_b = g_b(B). Random Variable with the points reduced for each B.
g_b = [0; -5; 0; -10];
% R_s = g_s(S). Random Variable with the points reduced for each S.
g_s = [0; 0; -3];

% The mean is linear.
E = E_Rh + E_Rb + E_Rs;

% Variance of a sum of three variableS.
% V = V_h + V_b + V_s + 2CovHB + 2CovHS + 2CovBS
E_HB = 0;
for h = 1:5
   for b = 1:4
       E_HB = E_HB + g_h(h) * g_b(b) * HB(h, b);
   end
end

E_HS = 0;
for h = 1:5
   for s = 1:3
       E_HS = E_HS + g_h(h) * g_s(s) * HS(h, s);
   end
end

E_BS = 0;
for b = 1:4
   for s = 1:3
       E_BS = E_BS + g_b(b) * g_s(s) * BS(b, s);
   end
end

CovHB = E_HB - E_Rh * E_Rb;
CovHS = E_HS - E_Rh * E_Rs;
CovBS = E_BS - E_Rb * E_Rs;

V = V_Rh + V_Rb + V_Rs + 2 * CovHB + 2 * CovHS + 2 * CovBS;

end