Page 113 - Probability and Statistical Inference

90 2. Expectations of Functions of Random Variables

2.2.6 Let X be a discrete random variable having the Geometric(p) distri-
bution whose pmf is f(x) = p(1 – p) for x = 1, 2, 3, ... and 0 < p < 1, given
x–1
by (1.7.7). Derive the explicit expressions of µ and σ for this distribution. An
alternative way to find µ has been provided in the Exercise 2.2.23. {Hint: Use
the expressions for sums of the second and third infinite series expansions
from (1.6.15).}
2.2.7 Suppose that X has the Poisson(λ) distribution with 0 < λ< ∞. Show
that µ = σ = λwhich would verify (2.2.18). See Section 2.2.3 for some
2
partial calculations.
2.2.8 For the negative binomial pmf defined by (1.7.10), find the expres-
sions for the mean and variance.
2.2.9 Let c be a positive constant and consider the pdf of a random variable
X given by

(i) Explicitly evaluate c, µ and σ;
(ii) Evaluate E{(1 – X) –1/2 };
(iii) Evaluate E{X (1 – X) }.
3/2
3
{Hints: In parts (ii)-(iii) first derive the expressions using direct integration
techniques. Alternatively reduce each expected value in parts (ii)-(iii) as a com-
bination of the appropriate beta functions defined in (1.6.25).}
2.2.10 Let c be a constant and consider the pdf of a random variable X given
by

Explicitly evaluate c, µ and σ.
2.2.11 Suppose that X has the Uniform(0, 1) distribution. Evaluate
(i) E[X –1/2 (1 – X) ];
10
(ii) E[X (1 – X) ];
5/2
3/2
(iii) E[(X + 2X – 3X )(1 – X) ].
1/2
5/2
10
2
{Hint: First express these expectations as the integrals using the Definition
2.2.3. Can these integrals be evaluated using the forms of beta integrals de-
fined in (1.6.25)?}
2.2.12 Consider a random variable X having the pdf f(x) = c(x – 2x + x )
2
3
I(0 < x < 1) where c is a positive number. Explicitly evaluate µ and σ for this
distribution. {Hint: In order to find c, should one match this f(x) with a beta
density?}