Page 117 - Probability and Statistical Inference

94 2. Expectations of Functions of Random Variables

(i) the mean µ is finite, but the variance σ is infinite;
2
(ii) the mean µ and variance σ are both finite, but µ is infinite;
2
3
(iii) µ is finite, but µ is infinite.
10 11
{Hint: Extend the ideas from the Example 2.3.2 and try other pdf’s similar
to that given in the “hint” for the Exercise 2.3.9.}
2.3.11 For the binomial and Poisson distributions, derive the third and
fourth central moments by the method of successive differentiation of their
respective mgf’s from (2.3.5) and (2.3.10).
2.3.12 Consider the expression of the mgf of an exponential distribution
given in (2.3.27). By successively differentiating the mgf, find the third and
fourth central moments of the exponential distribution.
2.3.13 Consider the expression of the mgf of a Chi-square distribution
given in (2.3.28). By successively differentiating the mgf, find the third and
fourth central moments of the Chi-square distribution.
2.3.14 (Exercise 2.2.16 Continued) Suppose that X has the lognormal pdf
for 0 < x < ∞, given by (1.7.27).
th
Show that the r moment η is finite and find its expression for each r = 1, 2,
r
3, ... . {Hint: Substitute y = log(x) in the integrals and see that the integrals
would resemble the mgf with respect to the normal pdf for some appropriate
values t.}
2.3.15 Suppose that a random variable X has the Laplace or the double
exponential pdf f(x) = 1/2β exp{– |x|/β} for all x ∈ ℜ where β ∈ ℜ . For this
+
distribution.
(i) derive the expression of its mgf;
(ii) derive the expressions of the third and fourth central moments.
{Hint: Look at the Section 2.2.6. Evaluate the relevant integrals along the
lines of (2.2.31)-(2.2.33).}
2.3.16 Suppose that Z has the standard normal distribution. Along the lines
of the Example 2.3.4, derive the expression of the mgf M (t) of the random
|Z|
variable |Z| for t belonging to ℜ.
2.3.17 Prove the Theorem 2.3.2.
2.3.18 (Exercise 2.2.14 Continued) Suppose that we have a random vari-
able X which has the Rayleigh distribution, that is its pdf is given by f(x) = 2θ –
r
1 xexp(–x /θ)I(x > 0) where θ(> 0). Evaluate E(X ) for any arbitrary but fixed
2
2
r > 0. {Hint: Try the substitution u = x /θ during the integration.}
2.3.19 (Exercise 2.2.15 Continued) Suppose that we have a random vari-
able X which has the Weibull distribution, that is its pdf is given by