If x y have distribution p sxycan be regarded as a random

Question

Continuing with the setup from the previous exercise we denote by P the probability measure with point probabilities, p(x, y), being the relative frequencies computed above. It is a probability measure on E₀ × E₀

• Compute the point probabilities, p₁(x) and p₂(y), for the marginal distributions of P and show that X and Y are not independent.
• Compute the score matrix defined as

Discuss the interpretation of the values.

• If (X, Y) have distribution P, S_X,Ycan be regarded as a random variable with values in a finite sample space (why?). Compute the mean of S_X,Y.
• Assume instead that X and Y are in fact independent with distributions given by the point probabilities p₁(x) and p₂(y) respectively and compute then the mean of S_X,Y.