Page 121 - Probability and Statistical Inference

98 2. Expectations of Functions of Random Variables

evaluate the k order factorial moment of X and show that it reduces to n(n –
th
1)...(n – k + 1)p .
k
2.5.5 Consider the pgf P (t) defined by (2.5.1). Suppose that P (t) is
X
X
twice differentiable and assume that the derivatives with respect to t and
expectation can be interchanged. Let us denote and
. Then, show that
(i)
(ii)
2.5.6 Suppose that a random variable X has the Beta(3, 5) distribution.
Evaluate the third and fourth factorial moments of X. {Hint: Observe that
E[X(X – 1)(X – 2)] = E[X(X – 1) – X(X – 1)] = E[X(X – 1) ] – E[X(X – 1)].
2
2
Write each expectation as a beta integral and evaluate the terms accordingly to
come up with the third factorial moment. The other part can be handled by
extending this idea.}