Consider a pattern classification problem where we would like to discriminate between two materials M_1 and M_2 using a measured color vector (R, G, B) of the unknown material. Assume that the a priori probability of M_1 is 3/7 and that the a priori probability of M_2 is 4/7. Suppose that the probability density for M_1 is uniform over the cube 50 lessthanorequalto R lessthanorequalto 90, 30 lessthanorequalto G lessthanorequalto 70, 40 lessthanorequalto B lessthanorequalto 80. Suppose that the probability density for M_2 is uniform over the cube 70 lessthanorequalto R lessthanorequalto 120, 30 lessthanorequalto G lessthanorequalto 80, 50 lessthanorequalto B lessthanorequalto 100.

A. What is the conditional pdf p(R, G, B|M_1) as a function of (R, G, B)?
B. What is the conditional pdf p(R, G, B|M_2) as a function of (R, G, B)?
C. What is the a posteriori probability P(M_1|R, G, B) as a function of (R, G, B)?
D. What is the a posteriori probability P(M_2|R, G, B) as a function of (R, G, B)?
E. What is the best guess for what material we are looking at as a function of (R, G, B)?
F. What is the probability of error as a function of (R, G, B) if we take the guess in part e)?