Consider the class of binary models (4.8) and (4.9). Suppose that the standard cdf D corresponds to a probability density function # that is symmetric around 0.
a. Show that x at which π(x) = 0.5 is x = – α/β.
b. Show that the rate of change in π(x) when π(x) = 0.5 is β∅(0). Show this is 0.25β for the logit link and β/V2π- (where it = 3.14 …) for the probit link.
c. Show that the probit regression curve has the shape of a normal cdf with mean —α/β and standard deviation 1/ |β |.