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2. Expectations of Functions of Random Variables 93

(i) E(Z ) = 0 for any odd integer r > 0;
r
2 Γ(1/2r + 1/2) for any even integer r > 0;
(ii) E(Z ) = π –1/2 r/2
r
(iii) µ = 3 using part (ii).
4
2.3.2 (Exercise 2.3.1 Continued) Suppose that a random variable X has the
N(µ, σ ) distribution. By directly evaluating the relevant integrals, show that
2
r
(i) E{(X – µ) } = 0 for all positive odd integer r;
r
r/2
(ii) E{(X – µ) } = π –1/2 σ 2 Γ (1/2r + 1/2) for all positive
r
even integer r;
(iii) the fourth central moment µ reduces to 3σ .
4
4
2.3.3 Suppose that Z has the standard normal distribution. Along the lines
of the Example 2.3.4, that is directly using the techniques of integration, show
that
for any number r > 0.

2.3.4 (Exercise 2.3.3 Continued) Why is it that the answer in the Exercise
r
2.3.1, part (ii) matches with the expression of E(|Z| ) given in the Exercise
2.3.3? Is it possible to somehow connect these two pieces?
2.3.5 (Exercise 2.2.13 Continued) Suppose that X is a continuous random
variable having its pdf f(x) with the support χ = (a, b), –∞ = a < b = ∞.
Additionally suppose that f(x) is symmetric about the point x = c where a < c
r
< b. Then, show that E{(X – c) } = 0 for all positive odd integer r as long as
the r order moment is finite. {Hint: In principle, follow along the derivations
th
in the part (i) in the Exercise 2.3.2.}
2.3.6 (Exercise 2.3.1 Continued) Derive the same results stated in the
Exercise 2.3.1 by successively differentiating the mgf of Z given by (2.3.14).
2.3.7 (Exercise 2.3.2 Continued) Suppose that X has the N(µ, σ ) distribu-
2
tion. Show that µ = 0 and µ = 3σ by successively differentiating the mgf of
4
3
4
X given by (2.3.16).
2.3.8 Consider the Cauchy random variable X, defined in (1.7.31), whose
pdf is given by f(x) = π (1 + x ) for x ∈ ℜ. Show that E(X) does not exist.
–1
2 –1
2.3.9 Give an example of a continuous random variable, other than Cauchy,
for which the mean µ is infinite. {Hint: Try a continuous version of the Ex-
ample 2.3.1. Is f(x) = x I(1 < x < ∞) a genuine pdf? What happens to the
–2
mean of the corresponding random variable?}
2.3.10 Give different examples of continuous random variables for which