Consider the class of binary models (4.8) and (4.9). Suppose that the standard cdf corresponds to a probability density function that is symmetric

Consider the class of binary models (4.8) and (4.9). Suppose that the standard cdf Φ corresponds to a probability density function ϕ 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 βϕ(0).

Show this is 0.25β for the logit link and β/√π (where π = 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|.