Page 52 - Elements of Distribution Theory

P1: JZP
052184472Xc01 CUNY148/Severini May 24, 2005 17:52

38 Properties of Probability Distributions

1.35 Let X denote a real-valued random variable with an absolutely continuous distribution with
density function
1 1
p(x) = √ exp − x 2 , −∞ < x < ∞.
(2π) 2

Let g : R → R denote a differentiable function such that E[|g (X)|] < ∞. Show that
E[g (X)] = E[Xg(X)].

1.10 Suggestions for Further Reading
The topics covered in this chapter are standard topics in probability theory and are covered in many
books on probability and statistics. See, for example, Ash (1972), Billingsley (1995), Karr (1993),
and Port (1994) for rigorous discussion of these topics. Capinski and Kopp (2004) has a particularly
accessible, yet rigorous, treatment of measure theory. Casella and Berger (2002), Ross (1995), Snell
(1988), and Woodroofe (1975) contain good introductory treatments. Theorem 1.11 is based on
Theorem 1.2 of Billingsley (1968).