For the population having value y on a binary response, suppose x has an N(????y, ????2) distribution, y = 0, 1.

a. Using Bayes’ theorem, show that P(y = 1 ∣ x) satisfies the logistic regression model with ????1 = (????1 − ????0)∕????2.

b. Suppose that (x ∣ y) ∼ N(????y, ????2 y ) with ????0 ≠ ????1. Show that the logistic model holds with a quadratic term (Anderson 1975).

c. Suppose that (x ∣ y) has natural exponential family density f(x; ????y) = h(x) exp[x????y − b(????y)].

Show that P(y = 1 ∣ x) satisfies the logistic model with ????1 = (????1 − ????0).