Suppose that (X, Y) is uniform on the set {(x1, y1),...,(xn, yn)} where the x1,..., xn are distinct values and the y1,..., yn are distinct values. (a) Prove that X is uniformly distributed on x1,..., xn, with mean given by x¯ = n−1 3n i=1 xi and variance given by sˆ2 X = n−1 3n i=1 (xi − ¯x) 2 . (b) Prove that the correlation coefficient between X and Y is given by rXY = 3n i=1 (xi − ¯x) (yi − ¯y) T3n i=1 (xi − ¯x) 2 T3n i=1 (yi − ¯y) 2 = sˆXY sˆX sˆY where sˆXY = n−1 3n i=1 (xi − ¯x) (yi − ¯y). The value sˆXY is referred to as the sample covariance and rXY is referred to as the sample correlation coefficient when the values (x1, y1),...,(xn, yn) are an observed sample from some bivariate distribution. (c) Argue that rXY is also the correlation coefficient between X and Y when we drop the assumption of distinctness for the xi and yi . (d) Prove that −1 ≤ rXY ≤ 1 and state the conditions under which rXY = ±1