Let f(x) be a pdf and let a be a number such that, for all ∈ > 0, f(a + ∈) = f(a - ∈). Such a pdf is said to be symmetric about the point a.
(a) Give three examples of symmetric pdfs.
(b) Show that if X ~ f(x), symmetric, then the median of X (see Exercise 2.17) is the number a.
(c) Show that if X ~ f(x), symmetric, and EX exists, then EX =a.
(d) Show that f(x) = e-x, x ≥ 0, is not a symmetric pdf.
(e) Show that for the pdf in part (d), the median is less than the mean.