Consider the choice between two options, such as two product brands. Let U₀ denote the utility of outcome y = 0 and U₁ the utility of y = 1. For y = 0 and 1, suppose that U_y = α_y + β_y x + ԑ_y, using a scale such that ԑ_y has some standardized distribution. A subject selects y = 1 if U₁ > U₀ for that subject.

a. If ԑ₀ and ԑ₁ are independent N(0, 1) random variables, show that P(Y = 1) satisfies the probit model.