The Face Recognition Technology (FERET) program, sponsored by the U.S. Department of Defense, was designed to develop automatic face recognition capabilities to assist homeland security. A biometric face “signature” of an unknown person (called the probe) is compared to a signature of a known individual from the “gallery” and a similarity score is measured. FERET includes algorithms for finding the gallery signature that best matches the probe signature by ranking similarity scores. In Chance (Winter 2004), the discrete Copula probability distribution was employed to compare algorithms. Let X represent the similarity score for a probe using algorithm A and Y represent the similarity score for the same probe using algorithm B. Suppose that the gallery contains signatures for n = 3 known individuals, numbered 1, 2, and 3. Then X₁, X₂, and X₃ represent the similarity scores using algorithm A and Y₁, Y₂, and Y₃ represent the similarity scores using algorithm B. Now rank the X values and define X_(i) so that X₍₁₎ > X₍₂₎ > X₍₃₎ . Similarly, rank the Y values and define Y_(i) so that Y₍₁₎ > Y₍₂₎ > Y₍₃₎ . Then, the Copula distribution is the joint probability distribution of the ranked X’s and Y’s, given as follows: 

P(X, Y) = 1/n if the pair (X_(x), Y_(y)) is in the sample, 0 if not where x = 1, 2, 3, ... n and y = 1, 2, 3,... , n. Suppose the similarity scores (measured on a 100-point scale) for a particular probe with n = 3 are (X₁ = 75, Y₁ = 60), (X₂ = 30, Y₂ = 80), (X₃ = 15, Y₃ = 5)and .

a. Give the Copula distribution, p(x, y), for this probe in table form.

b. Demonstrate that if both algorithms agree completely on the signature match, then p(1, 1) = p(2, 2) = p(3, 3) = 1/3.