Let # be the standard normal distribution function (mean 0, variance 1). Let = # be

Let # be the standard normal distribution function (mean 0, variance 1).

Let φ = # be the density. Show that φ

(x) = −xφ(x). If x > 0, show that

 ∞

x zφ(z) dz = φ(x) and 1 − #(x) <  ∞

x z

x

φ(z) dz.

Conclude that 1 − #(x) < φ(x)/x for x > 0. If x < 0, show that

#(x) < φ(x)/|x|. Show that log # and log(1−#) are strictly concave, because their second derivatives are strictly negative. Hint: do the cases x > 0 and x < 0 separately.