Page 120 - Probability and Statistical Inference

2. Expectations of Functions of Random Variables 97

Derive the explicit expression of the mgf of X and hence derive µ and σ for
n
this distribution. {Hint: Note that E(e ) = Σ et 1/n = 1/n e [1+e +
1
t
tx
t
i=1
. Can this be simplified using the geometric progression?}
2.4.8 A random variable X has its pdf
where c(> 0) and d are appropriate constants. We are also told that E(X) =
–3/2.
(i) Determine c and d;
(ii) Identify the distribution by its name;
(iii) Evaluate the third and fourth central moments of X.
2.4.9 A random variable X has its pdf

where c(> 0) and d(> 0) are appropriate constants. We are also told that E(X)
= 10.5 and V(X) = 31.5.
(i) Determine c and d;
(ii) Identify the distribution by its name;
(iii) Evaluate the third and fourth moments, η and η , for X.
3 4
2.4.10 A random variable X has its mgf given by

Evaluate P(X = 4 or 5). {Hint: What is the mgf of a geometric random vari-
able?}
2.5.1 Verify (2.5.4).
2.5.2 Let X be a discrete random variable having the Geometric(p) distri-
bution whose pmf is f(x) = P(X = x) = p(1 – p) for x = 1, 2, 3, ... and 0 <
x–1
p < 1, given by (1.7.7). Show that pgf of X is given by

Evaluate the first two factorial moments of X and hence find the expressions
of µ and σ.
2.5.3 Let X be a discrete random variable having the Poisson(λ) distribu-
tion with 0 < λ < ∞. Derive the expression of the pgf of X. Hence, evaluate the
k order factorial moment of X and show that it reduces to λ .
th
k
2.5.4 Let X be a discrete random variable having the Binomial(n, p)
distribution with 0 < p < 1. Derive the expression of the pgf of X. Hence,