Let f(x) be a pdf, and let a be a number such that if a ≥ x ≥ y, then f(a) ≥ f(x) ≥ f(y), and if a ≤ x ≤ y, then f(a) ≥ f(x) ≥ f(y). Such a pdf is called unimodal with a mode equal to a.
(a) Give an example of a unimodal pdf for which the mode is unique.
(b) Give an example of a unimodal pdf for which the mode is not unique.
(c) Show that if f(x) is both symmetric (see Exercise 2.26) and unimodal, then the point of symmetry is a mode.
(d) Consider the pdf f(x) = e-x, x ≥ 0. Show that this pdf is unimodal. What is its mode?